[i|At>/nON\ 

a  Jgxt  Book, 

-H  A- Clarke 


Digitized  by  the  Internet  Archive 
in  2014 


https://archive.org/details/systemofharmonyfOOclar_0 


A  SYSTEM 

or 

HARHONY 


POUNDED  ON  KEY  RELATIONSHIP 


By  means  of  which  a  thorough  Knowledge  of 
the  rules  that  govern  the  combinations  and 
successions  of  sounds  may  be  easily  acquired 


WITH  OR  WITHOUT  A  TEACHER 


H.  T\.  CL3RKE,  MUS.  DOC. 

pr-joressoi?  or  nusic  in  the  university  or  Pennsylvania 


^rjiladelprjict- 


Copyright,  1898,  by  Theo.  PRESSER 


PREFACE. 


This  work  is  not  meant  to  be  a  theory  of  harmony,  but  a  simple 
practical  system,  by  means  of  which  a  knowledge  of  the  mass  of  facts 
that  form  the  basis  of  the  art  of  composition  may  be  acquired.  Theo- 
retical questions  are  therefore  carefully  avoided,  and  the  tempered 
scale  is  accepted  as  the  foundation  of  modern  music. 

The  principal  assumption  or  hypothesis  upon  which  the  system  is 
built  is  the  following  : 

No  scale  is  an  independent  entity,  but  is  only  the  principal  one 
in  a  group  of  six  called  the  related  group. 

The  other  is  the  reference  of  all  dissonant  groups  to  two  roots  in 
each  scale. 

The  related  group  is  of  course  familiar  to  all  musicians,  but  it  has 
never  been  used  as  the  basis  of  a  system  of  teaching  harmony,  to  any 
great  extent.  The  derivation  of  dissonarit  groups  from  a  few  roots 
is  also  quite  familiar  now,  yet  some  departures  from  the  usual  manner 
of  treating  them  will  be  found  in  this  work,  not  for  theoretical  rea- 
sons, but  for  the  gain  of  greater  simplicity. 

The  plan  of  teaching  laid  down  in  this  work  is  entirely  new  in 
several  respects,  and  is  based  on  the  following  maxims : 

Teach  one  thing  at  a  time. 

Arrange  the  subject  matter  in  such  way  that  each  step  is  a  natural 
outgrowth  of  the  last. 

Teach  the  pupil  how  to  use  his  knowledge,  instead  of  simply  try- 
ing to  follow  out  the  rule.  Another  departure  from  the  usual  prac- 
tice is  the  complete  discarding  of  the  figured  bass,  with  the  result 
that  all  merely  mechanical  rotework  is  eliminated,  and  the  pupil 
writes  the  chords  because  he  knows  what  they  are. 

Many  of  the  rules  usually  found  in  text-books  are  omitted ;  many 
of  them  are  modified.    For  the  reason  that  the  so-called  rules  of 


PREFACE. 


composition  have  all  been  arrived  at  empirically,  they  are  in  a  great 
measure,  and  ought  to  be  altogether,  deductions  from  the  practice  of 
those  best  qualified  to  say  what  should  or  should  not  be;  viz.,  the 
great  composers.  It  is  a  common  subject  of  complaint  on  the  part 
of  the  student  that  such  and  such  things  are  laid  down  in  the  text- 
books as  rules  that  must  not  be  broken,  and  when  he  turns  to  the 
work  of  some  great  composer  he  finds  them  totally  disregarded.  His 
complaint  is  generally  met  by  the  reply,  "When  you  can  write  like 
so  and  so,  you  may  break  the  rules  also."  A  wiser  reply  would  be, 
Whatever  has  been  sanctioned  by  a  great  writer  becomes  the  property 
of  all,  provided  they  can  use  it  with  taste  and  discretion. 

We  do  not  wish  to  be  understood  as  recommending  to  the  pupil 
the  instant  disregard  of  all  rules,  because  no  one  can  learn  how  to 
break  them  until  he  has  learned  thoroughly  well  how  to  use  them. 

It  being  the  business  of  harmony  to  teach  the  combinations  and 
succession  of  sounds,  very  little  is  said  in  this  book  about  the  move- 
ment of  parts,  that  being  the  province  of  counterpoint,  which  may 
be  defined  as  the  art  of  making  the  best  use  of  the  materials  placed 
at  our  disposal  by  harmony. 

H.  A.  CLARKE,  Mus.  D. 
Professor  of  Music,  University  of  Penna. 

January,  1898. 


CONTENTS. 


I NTRODUCT70N  PAGE  V 

Chapter  I.  page  i 

Intervals. 

Chapter  II.  page  6 

Major  Scale. 

Chapter  III.  page  q 

Major  Scale  and  Common  Chord. 

Chapter  IV.  page  13 

Positions  of  Common  Chord  and  Succession  of  Common  Chords 
with  Roots  at  Bass. 

Chapter  V.  page  16 

Sequences  of  Common  Chords. 

Chapter  VI.  page  22 

First  Inversion  of  Common  Chord. 

Chapter  VII.  page  27 

Second  Inversion  of  Common  Chord. 

Chapter  VIII.  page  30 

Harmonizing  of  Melodies  with  Common  Chords  and  their  In- 
versions. 

Chapter  IX.  page  33 

The  Minor  Scale. 

Chapter  X.  page  38 

The  Group  or  Circle  of  Related  Keys. 

Chapter  XI.  page  43 

Chords  of  Parallel  Minor,  Lowered  Supertonic, —  and  Chords  in 
the  Related  Keys  not  Found  in  the  Given  Key. 

Chapter  XII.  page  51 

Chord  of  Dominant  Seventh, —  First  Progression. 

Chapter  XIII.  page  56 

Chord  of  Dominant  Seventh, —  Second  and  Third  Progressions 
and  Succession. 


CONTENTS 


Chapter  XIV.  page  60 

Chord  of  Dominant  Ninth. 

Chapter  XV.  page  6i 

Chord  of  Dominant  ElevCiVtn. 

Chapter  XVI.  .    page  72 

Chord  of  Dominant  Eleventh  Continued, —  Additional  Remancs 
on  Second  Inversions. 

Chapter  XVII.  page  76 

Progression  of  Dissonant  Chords  by  means  of  Related  Sounds. 

Chapter  XVIII.  page  83 

Chord  of  Thirteenth. 

Chapter  XIX.  page  87 

Supertonic  Harmony. 

Chapter  XX.  page  91 

Altered   Chords,  —  Augmented   Sixth,  —  Augmented   P'ifth, — 
Passing  Seventh. 

Chapter  XXI.  page  97 

Suspensions. 

Chapter  XXII.  page  99 

Retardation. 

Chapter  XXIII.  page  109 

Changing  Notes  or  Appoggiaturas. 

Chapter  XXIV.  page  112 

Passing  Notes. 

Chapter  XXV.  page  119 

Minor  Scale. 

Chapter  XXVI.  page  123 

Open  or  Vocal  Harmony. 

Chapter  XXVII.  page  126 

Pedal  or  Organ  Point. 

Chapter  XXVIII.  page  130 

Transition. 


Supplement. 

Tempered  Scale, —  Figured  Bass. 


page  145 


INTRODUCTION. 


Sound  is  produced  by  the  motion  of  the  air.  This  motion  is  com- 
municated to  it  by  the  vibration  of  some  material  body.  If  the  vibra- 
tions are  irregular,  the  sound  produced  is  called  Noise  ;  if  they  are 
regular,  the  sound  produced  is  called  Musical. 

The  means  used  for  producing  musical  sounds  for  artistic  purposes 
are : 

ist.  The  vibration  of  a  Column  of  Air  enclosed  in  a  tube,  as  in  the 
flute  and  the  flue-pipes  of  the  organ. 

2d.  The  combination  of  the  column  of  air  with  a  Reed,  as  in  the 
clarinet,  hautboy,  and  ( in  some  degree  )  the  human  voice. 

3d.  The  vibration  of  Strings  set  in  motion  by  the  fingers,  or  by  a 
bow,  or  by  hammers,  as  in  the  harp,  violin,  and  piano.  The  brass  in- 
struments may  be  included  among  the  reed  instruments,  as  the  lip  of 
the  player  in  this  case  performs  the  function  of  the  reed. 

4th.  The  vibration  of  the  free  reed,  as  in  the  harmonium. 

Musical  sounds  are  distinguished  among  themselves  in  various 
ways  : 

ISt.   As  high  or  low  (  acute  or  grave  )  ,  called  Pitch. 

2d.  Loud  and  soft  (forte  and  piano),  called  Intensity. 

3d.  Quality  (timbre).  By  quality  is  meant  that  property  of  a 
musical  sound  by  which  we  distinguish  whether  it  is  produced  by  a 
violin,  or  flute,  or  voice,  etc. 

4th.  By  the  relative  length  of  time  the  sounds  last,  called  Du- 
ration. 

Notation  is  a  system  of  signs  designed  to  represent  the  Pitch,  Du- 
ration, and  Intensity  of  sounds. 

The  signs  used  to  designate  Pitch  are  the  Staff  and  the  Clefs;  to 
designate  Duration,  the  Notes  ; 

(v) 


vi 


IN  TROD  UC  TION. 


To  designate  Pitch  and  Duration  together,  the  Position  of  the 
Notes  on  the  Staff  and  their  Form. 

The  Staff  consists  of  five  parallel  lines,  on,  between,  and  above  or 
below  which  the  notes  are  placed.  When  the  staff  is  insufficient,  addi- 
tional lines  called  Leger  Lines  ( generally  but  erroneously  written 
ledger  )  are  used,  and  the  notes  are  written  on,  between,  above,  or 
below  them. 

The  Clefs,  three  in  number,  are  placed  at  the  beginning  of  the  staff  ; 
they  determine  the  letter  name  of  the  line  upon  which  they  are  placed. 

ist.  The  treble  or  G  clef,  used  now  only  on  the  second  line. 
2d.  The  C  clef,  Sr  or         used  on  the  first  line,  is  called  the 


soprano  or  descant  clef ;  on  the  third  line,  called  the  Alto  Clef  ;  and 
on  the  fourth  line,  called  the  Tenor  Clef.  The  use  of  this  clef  on  the 
first  line  is  gradually  being  discontinued. 

3d.  The  Bass  or  F  clef,  CJj,  used  now  only  on  the  fourth  line, 

was  formerly  used  on  the  third  line  also,  and  then  called  the  baritone 
clef. 

The  Pitch  of  the  G  indicated  by  is  that  of  the  sound  given  by 
an  open  pipe  sixteen  inches  long ;  of  the  C  indicated  by  that  of 
one  twenty-four  inches  long ;  of  the  F  indicated  by  ^f,  that  of  one 

thirty-six  inches  long. 

In  vocal  music  the  use  of  the  G  clef  for  the  tenor  voice  has  become 
almost  universal.  When  so  used  its  Pitch  must  be  understood  as 
being  an  octave  lower;  viz.,  that  of  the  sound  produced  by  a  pipe 
thirty-two  inches  long. 

The  Forms  of  notes  indicate  their  relative  duration  ;  their  Positions 
on  the  staff  their  letter  names. 

Eight  forms  of  notes  are  used  ;  the  duration  of  each  is  half  that  of 
the  one  that  precedes  it.  The  first  and  last  forms  are  less  used 
than  the  others ;  they  are  named : 

Breve,  or         Whole    Half    Quarter   Eighth   Sixteenth   Thirty-sec-  Sixty-fourth 
double  whole  note.    note.     note.     note.       note.       note.        ond  note.  note. 

iM!    -  f  r  Z  %    %  % 


INTRODUCTION. 


vn 


When  eighth,  sixteenth,  thirty-second,  and  sixty-fourth  notes  are 

0  0-    #■  0    0-  0-    0  0- 

used  in  groups  they  are  joined,  thus,  —     tZH  — -  in  instru- 

mental music,  but  not  in  vocal,  unless  they  are  all  sung  to  the  same 
syllable,  when  they  must  also  be  enclosed  in  a  line  called  a  slur; 

thus  :  F9^=^zj^ij:j  When  sung  to  separate  syllables  they  must  not 


fair 


be  joined ;  thus  :  f — * — J — 


sup  -  pli  -  ca  -  tion 

A  dot  placed  after  a  note,  thus,  ^  •  increases  its  duration  by  one- 
half  of  itself,  making  it  equal  to  three  of  the  note  that  follows  it  in  the 
table  above,  thus,  <s>  .  —  &    &   &  •  a  second  dot  increases  the  duration 

by  one-half  of  the  first  dot,  thus,  <s>  . .  —  &    ^  & 

Each  note  has  an  equivalent  sign  for  silence,  called  a  Rest,  thus : 

Whole      Half    Quarter      Eighth      Sixteenth      Thirty-second  Sixty-fourth 
rest.        rest.       rests.         rest.  rest.  rest.  rest. 

i  *  r      i  3 

Dots  modify  the  duration  of  rests  in  the  same  way  as  that  of  notes. 

The  essential  rhythm  of  music  is  indicated  by  dividing  it  into  por- 
tions of  equal  duration  by  lines  drawn  across  the  staff,  called  Bars. 
The  enclosed  spaces  are  called  Measures,  but  in  ordinary  speech  are 
also  called  bars. 

To  indicate  the  rhythmic  content  of  the  measure,  signs  called 
Time  Signatures  are  placed  after  the  clef ;  these  are  the  following : 

ijz    E  or  |         called  simple  common  time. 

£1  il  ii 

S  ^  §    Ifl'  ca^e<^  simple  triple  time. 

8  §  M    § '  ca^ec^  compound  common  time. 

^  g  g»'   called  compound  triple  time. 


viii 


INTRODUCTION. 


The  lower  figure  indicates  the  note  taken  as  the  unit  of  the  beat  or 
pulse  \  the  upper  figure,  the  number  of  these  units  that  each  measure 
must  contain. 

The  first  two  signs,  and  g,  are  survivals  of  the  old  sign  for 
what  was  called  imperfect  time  ;  viz.,  a  broken  circle,  which  signified 
two  beats  in  the  measure.  The  first  of  these  signs,  |Jj,  always  means 
two  beats  in  the  measure,  and  is  called  alia  breve  time ;  the  g  with- 
out  the  line  is  the  same  as       i.  e.,  four  beats  in  the  measure. 

The  first  four  are  called  Simple  Common  Time.  Common  time 
means  an  even  number  of  beats  in  a  measure  ;   simple  means  that 

a  single  rhythmic  unit  is  contained  in  each  measure,  (g  or   ^  is, 

strictly  speaking,  a  compound  of  two  rhythmic  units,  but  custom  includes  it  among  the  simple 
times.) 

The  second  four  are  called  Simple  Triple  Time  because  each  meas- 
ure contains  three  beats,  or  one  rhythmic  unit  of  1,  2,  3. 

The  Compound  Times  are  all  made  from  Simple  Triple  Time.  If 

the  upper  figure  is  6,  it  signifies  that  two  measures  of  simple  triple 
are  made  into  one;  if  the  upper  figure  is  12,  four  measures  of  simple 
triple  are  made  into  one.  Then,  as  6  and  12  are  even  numbers,  it  is 
called  Compound  Common  Time. 

The  last  four  consist  of  three  measures  of  triple  time  made  into 
one;  and  as  the  upper  figure  is  the  odd  number  9,  it  is  called  Com- 
pound Triple  Time. 

A  Scale  is  a  succession  of  sounds  ascending  or  descending  from  a 
given  sound.  There  are  two  kinds  of  scale,  Diatonic  and  Chromatic. 
A  Diatonic  Scale  moves  through  the  letters  without  omission  or 
repetition.  A  Chromatic  Scale  repeats  the  letters,  changing  their 
pitch  by  means  of  signs  called  sharp  ($),  flat  (b),  natural  (fc[),  double 
sharp  (x),  double  flat  (bb). 

The  interval  between  a  given  letter  and  the  next  above  or  below  is 
called  a  whole  tone  if  another  sound  comes  between  them,  and  a  half 
tone  if  no  sound  comes  between  them.  Natural  half-tones  are  found 
between  E  and  J?,  and  B  and  C;  in  every  other  case,  if  a  half-tone 
is  desired  between  two  letters,  one  or  the  other  must  be  modified  in 
pitch  by  means  of  a  sharp  or  flat. 


IN  TR  ODUC  TJON. 


IX 


The  sharp  raises  the  pitch  of  a  letter ;  the  flat  lowers  the  pitch  of 
a  letter.    Therefore,  to  make  a  half-tone  between  F  and  G,  the  F 


There  are  two  forms  of  the  Diatonic  Scale.  First,  called  the  Major 
Scale,  has  the  sounds  so  arranged  that  there  is  a  half-tone  between  the 
third  and  fourth,  and  seventh  and  eighth.  If  the  letter  C  is  chosen  as 
the  starting  note  of  a  scale  (called  the  Keynote),  we  get  a  scale  of 
this  kind  without  having  to  use  a  sharp  or  flat;  therefore  the  scale  of 
C  is  called  the  Natural  Major  Scale. 

If  we  take  the  sounds  of  the  scale  of  C  and  rearrange  them,  begin- 
ning with  A  as  a  keynote,  we  get  the  other  form  of  diatonic  scale 
called  the  Natural  Minor  Scale. 

The  chief  difference  between  a  major  and  minor  scale  is,  that  in 
the  major  there  are  two  whole-tones  from  the  keynote  to  the  third 
letter ;  in  the  minor,  a  tone-and-a-half  from  the  keynote  to  the  third 
letter. 

As  the  scale  of  A  minor  is  a  rearrangement  of  the  sounds  of  the 
scale  of  C  major,  it  is  called  the  Relative  Minor  of  C. 

If  a  scale  is  begun  on  any  other  letter  as  a  keynote,  it  is  necessary 
to  use  sharps  or  flats  to  make  the  whole  and  half-tones  fall  in  the 
proper  places.  When  a  piece  of  music  is  written  in  any  of  these 
scales,  the  sharps  or  flats  that  the  scale  requires  are  put  at  the  begin- 
ning, and  are  called  the  Signature.  Sharps  and  flats  to  the  number 
of  six  may  be  used  in  a  signature.    These  signatures  are  here  given : 


must  be  raised  or  the  G  must  be  lowered,  thus  : 


P- 


Keys  with  Sharp  Signatures. 


i . 


1.  Key  of  G,  or  of  its  relative  minor,  F  minor. 

2.  Key  of  D,  or  of  its  relative  minor,  JB  minor. 

3.  Key  of  A,  or  of  its  relative  minor,  F$  minor. 

4.  Key  of  F,  or  of  its  relative  minor,  C$  minor. 

5.  Key  of  B,  or  of  its  relative  minor,  G%  minor. 

6.  Key  of  F\,  or  of  its  relative  minor,  D$  minor, 


2. 


INTR  OD  UC  TION. 
Keys  with  Flat  Signatures. 
3-  4-  5- 


& — 1" 

1 .  Key  of  F or  of  its  relative  minor,  Z>  minor. 

2.  Key  of  .Z?b,  or  of  its  relative  minor,  G  minor. 

3.  Key  of         or  of  its  relative  minor,  C  minor. 

4.  Key  of  Ab,  or  of  its  relative  minor,  F  minor. 

5.  Key  of  Z>1?,  or  of  its  relative  minor,  B\>  minor. 

6.  Key  of  Gb,  or  of  its  relative  minor,  F\>  minor. 

Observe  that  the  sharps  always  begin  with  F§,  and  that  they  are  al- 
ways written  in  the  order  in  which  they  are  here  set  down.  So  also 
with  the  flats,  which  always  begin  with  B\> ;  also,  that  F§  and  G\}  are 
identical  in  pitch. 

If  the  number  of  flats  be  increased  to  seven,  the  keynote  would 
be  Cb,  this  being  the  same  as  F,  which  has  five  sharps.  It  is  preferable 
to  write  in  F,  it  being  easier  to  read  in  the  key  which  has  the  fewer 
modified  letters.  Keys  with  more  than  six  sharps  or  flats  often  occur 
in  the  course  of  a  piece  of  music,  but  the  sharps  and  flats  are  then 
placed  before  the  letters  to  which  they  belong. 

When  any  sharp  01  flat,  not  found  in  the  signature,  occurs  in  the 
course  of  a  piece  of  music,  it  is  called  an  Accidental. 

Accidentals  affect  all  the  letters  on  the  same  degree  on  which  they 
are  written,  but  their  influence  never  extends  beyond  the  measure  in 
which  they  occur,  except  when  they  occur  at  the  end  of  a  measure  and 
the  same  letter  is  repeated  at  the  beginning  of  the  next  measure. 

(This  rule  is  very  generally  disregarded  now,  and  the  accidental  is  repeated.) 

The  Natural  is  never  used  in  the  signature,  except  when  in  the 
course  of  a  piece  a  change  of  signature  occurs,  when  the  old  signa- 

ture  is  canceled,  thus  : 


When    the  natural 


occurs  in  the  course  of  a  piece  to  cancel  one  of  the  sharps  or  flats  in 
the  signature,  it  is  an  accidental.  But  when  it  occurs  for  the  purpose 
of  restoring  an  altered  note  to  its  place  in  the  scale,  it  is  not  an  acci- 
dental. 


INTR  OD  UC  TION. 


xi 


The  natural  cancels  the  double  sharp  and  double  flat  also.  Therefore, 
if  the  double  sharped  letter  is  to  be  restored  to  its  original  place,  the 
natural  and  sharp  are  combined,  thus,  t$ ;  if  the  letter  is  double  flat, 
thus,  t[b.  The  following  illustration  gives  examples  of  all  the  subjects 
treated  so  far. 


In  piano  music  two  staves  are  used,  joined  with  a  sign  called  a 
Brace.  Immediately  after  the  brace  the  Clefs  are  placed ;  then  the 
Signature,  then  the  Time  Signature.  In  the  second  measure  the 
natural  is  an  accidental.  A  quarter  and  a  half -rest  are  also  found 
in  this  measure.  After  the  fourth  measure  the  signature  is  changed  — 
also  the  time  signature.  Whenever  either  one  of  these  is  changed  a 
Double  Bar  must  precede  the  change. 

The  clefs  and  signature  must  be  put  at  the  beginning  of  every  line, 
but  not  the  time  signature.  Observe  that  in  every  measure  the 
united  duration  of  the  notes  and  rests  equals  the  rhythmic  unit  indi- 
cated by  the  Time  signature. 

In  the  eighth  measure  a  whole-measure  rest  is  indicated  in  the 
lower  part.  A  whole-rest  is  always  used  for  this  purpose,  without 
regard  to  what  the  time  signature  may  be. 


xii 


INTR  OD  UC  TION. 


The  duration  of  a  note  may  be  varied  in  another  way;  viz.,  by 
tying  it  to  another  or  to  several  more  on  the  same  degree.  This  sign 
- — ^  is  called  a  tie.    The  tie  must  be  repeated  for  each  note  when 

several  are  tied,  thus :  j5*     ^     ^     j5*     j*  J 

If  the  first  note  is  marked  with  an  accidental,  it  is  not  necessary 
to  repeat  it  as  long  as  the  tie  lasts. 

The  duration  of  a  note  is  also  indefinitely  extended  by  means  of  a 
sign  called  a  pause  or  fermata         The  pitch  of  the  G  clef  is  often 

raised  an  octave  by  writing  8va   over  it.    The  octave  higher 

lasts  until  the  dotted  line  ceases. 

The  pitch  of  the  F  clef  may  be  lowered  an  octave  by  writing  under 

it  8vb   Sva  is  an  abbreviation  of    ottava  alt  a  (octave 

higher)  ;  8vb  is  an  abbreviation  of  ottava  bassa  (octave  lower) . 

The'  Rate  of  Movement,  called  Tempo,  is  indicated  by  Italian 
words,  as,  allegro,  andante,  etc.  ; 

The  Intensity,  by  p,  pft,  ppp,  which  stand  for  piano  (soft) ,  piu 
piano  (softer),  pianissimo  (softest),  fiffifff,  which  stand  for  forte 
(loud),  piu  forte  (louder),  fortissimo  (loudest)  ;  sudden  access  of  in- 
tensity, by  or  a,  ox  fz\  gradual  increase  of  intensity,  -===;  grad- 
ual decrease,  z==-.  Many  other  words  and  signs  are  used,  but  they 
may  all  be  found  in  the  Dictionary  of  Musical  Terms. 

Rules  for  writing  the  chromatic  scale  in  any  given  key  :  In  ascend- 
ing, raise  all  the  letters  except  the  third  and  seventh. 

In  descending,  lower  all  the  letters  except  the  first,  fifth,  and 
fourth. 

When  any  one  of  the  major  chords  in  the  scale  accompanies  it, 
think  of  it  as  a  tonic  chord  and  write  accordingly. 

Write  in  the  same  way,  if  the  accompanying  chord  is  its  domi- 
nant. The  notes  that  must  be  changed  in  accordance  with  this  rule 
are  marked  x. 


INTRODUCTION-. 


xiii 


of  AJf. 

FJt. 

1  V 

-w-J 

9± 


:or: 


Mi 


Ctf ,  not 

Db. 


-S: 


5  th  4th 


If  the  accompanying  chord  is  one  of  the  minor  to?iics,  or  its  dom- 
inant, write  as  though  in  that  key,  as  follows  :  in  ascending,  raise 
all  but  the  second  and  fifth  ;  descending,  lower  all  but  the  first  and 
fifth. 


->5>- 
-'5'- 


9£ 


m 


Ml 


xiv 


INTRODUCTION. 


The  advantages  gained  by  this  method  of  writing  the  chromatic 
scale  are  two. 

i  st.  It  reduces  the  accidentals  to  the  smallest  number  possible. 

2d.  It  does  not  introduce  an  accidental  that  may  not  be  found  in 
the  dominant  harmonies  of  the  related  group  with  one  exception,  viz., 
A$\  and  even  this  is  possible  as  the  augmented  fifth  in  the  dominant 
of  G,  and  as  the  third  in  the  supertonic  harmony  of  its  relative  minor, 
E. 

DEFINITIONS. 

Motion  is  similar  when  two  parts  or  voices  ascend  or  descend  to- 
gether ;  oblique  when  one  part  is  stationary  while  the  other  moves ; 
co7ttrary,  when  the  parts  move  in  opposite  directions. 

A  degree  is  from  one  letter  to  the  next  above  or  below,  whether 
distant  a  whole-tone  or  half-tone.  Conjunct  motion  is  motion  by  de- 
grees ;  disjunct  motion  is  motion  by  leaps ;  diatonic  motion  is  from 
one  letter  to  the  next;  chromatic  motion  is  from  any  letter  to  an 
altered  form  of  the  same  letter ;  as,  A-A\>  or  A-A\. 

Enharmonic  change  is  the  substitution  of  one  letter  for  another 
without  changing  the  pitch ;  as,  Cjf,  D^. 


HARMONY. 


CHAPTER  I. 
Intervals. 

Harmony  treats  of  the  combinations  of  sounds  of  different  pitch, 
and  the  successions  of  these  combinations. 

The  basis  of  modern  music  is  a  series  of  sounds,  each  one  of 
which  differs  from  the  one  lying  next  above  or  below  by  an  interval 
called  a  half-tone,  semitone,  half-step,  or  minor  second. 

If  the  two  contiguous  sounds  are  expressed  by  the  same  letter,  as 
C,  Cjf.  B,£b.  it  is  called  a  Chromatic  half-tone;  if  by  different 
letters  it  is  called  a  Diatonic  half-tone. 

When  the  same  sound  is  represented  by  different  letters,  as  C$, 
Z>b,  it  is  called  an  Enharmonic  change. 

The  difference  in  pitch  between  two  sounds  is  called  an  Interval. 

The  number  of  letters  included  decides  the  name  of  the  interval, 
without  regard"  to  the  number  of  whole  or  half-tones  it  may  include. 
Thus,  C-D$  is  called  a  second  because  it  includes  two  letters, 
while  C—E  t>,  which  sounds  the  same  ( in  the  modern  scale ) ,  is 
called  a  third  because  it  includes  three  letters. 

The  names  of  the  intervals  within  the  limits  of  the  octave  are  : 
Second,  Third,  Fourth,  Fifth,  Sixth,  Seventh,  Octave. 

But  every  interval  may  be  written  in  several  ways;  as,  C-T)\>, 
C—D,  C-D\.  It  is  therefore  necessary  to  distinguish  between  the 
different  kinds  of  interval  bearing  the  same  name. 

This  is  done  by  the  use  of  the  following  terms,  added  to  the 
name  to  denote  the  kind :  Minor,  or  small ;  Major,  or  large ;  Per- 
fect, Diminished,  and  Augmented. 

Intervals  are  also  classified  as  Consonant  and  Dissonant.  A  con- 
sonant interval  is  one  that  gives  repose  to  the  ear.    A  dissonant  in- 

(i) 


2 


HARMONY. 


terval  is  one  of  which  it  is  necessary  that  one  or  both  the  sounds 
must  move  in  a  certain  way  to  satisfy  the  ear. 

Consonant  intervals  are  farther  divided  into  Perfect  and  Imperfect. 

A  perfect  consonance  is  one  that  cannot  be  altered  without  produc- 
ing a  dissonance.  An  imperfect  consonance  is  one  that  is  equally 
consonant,  whether  major  or  minor. 

A  diminished  interval  results  from  the  contraction  of  a  minor  or 
perfect  interval. 

An  augmented  interval  results  from  the  expansion  of  a  major  or 
perfect  interval. 

All  aug.nented  and  diminished  intervals  are  dissonant. 

The  motion  of  the  member  or  members  of  a  dissonant  interval  is 
called  their  resolution. 

If  the  dissonance  is  minor  or  major,  only  one  member  is  compelled 
to  move  ;  if  it  is  augmented  or  diminished,  both  members  must  move, 
towards  each  other  if  the  interval  is  diminished,  away  from  each 
other  if  it  is  augmented. 

Although  there  are  five  kinds  of  interval,  there  are  not  five  kinds 
of  every  interval.    Thus  of  seconds  there  are  three  kinds,  viz.  : 
C-D\>,  one  half-tone,  called  minor  second. 
C-D,  two  half-tones,  called  major  second. 
C-Z>$,  three  half-tones,  called  augmented  second. 

It  is  quite  possible  to  put  a  diminished  second,  or  a  doubly  aug- 
mented second  on  paper ;  but  as  neither  are  to  be  found  in  any  possi- 
ble combination  or  succession,  they  are  excluded  as  of  no  practical 
use. 

The  third  also  exists  in  three  forms,  viz.  : 
Cj-jE'b,  diminished,  two  half-tones. 
C—Jlb,  minor,  three  half-tones. 
C-E,  major,  four  half-tones. 
The  fourth  exists  in  three  forms,  viz.  : 
C\-F,  diminished,  four  half-tones. 
C-F,  perfect,  five  half-tones. 
C-jFJ},  augmented,  six  half-tones. 

The  remaining  intervals  within  the  limits  of  the  octave  may  be 
found  by'inverting  the  letters  of  those  just  given.  It  will  at  once  be 
evident  that  any  interval  and  its  inversion  must  together  make  an  oc- 


HARMONY.  3 

2d.  7th. 

tave,  thus,   C    D        C;  and  since  twelve  half-tones  make  an  octave, 

an  interval  and  its  inversion  must  make  twelve  half-tones.  There- 
fore, to  find  the  number  of  half-tones  in  the  inversion  of  a  given  in- 
terval, it  is  necessary  only  to  subtract  the  number  in  the  interval  from 
twelve,  the  number  in  the  octave  ;  thus,  in  the  second  C-D  there  are  two 
half-tones,  therefore  there  must  be  ten  in  its  inversion,  D-C\  and  as  the 
number  of  letters  from  D  to  C  is  seven,  we  also  find  that  the  inversion 
of  a  second  produces  a  seventh.  Then  the  inversion  of  a  third  must 
produce  a  sixth,  and  the  inversion  of  a  fourth  must  produce  a  fifth. 

Then,  as  there  are  three  kinds  of  each  interval,  three  kinds  of 
sevenths,  sixths,  and  fifths  must  result  from  their  inversion ;  and  the 
nearer  each  other  the  sounds  are  in  the  smaller  interval,  the  farther 
apart  they  must  be  in  its  inversion.  Therefore  the  inversion  of  a 
given  interval  always  produces  one  of  the  opposite  kind,  with  one  ex- 
ception; viz.,  the  inversion  of  a  perfect  interval  produces  a  fieifect 
one. 

The  foregoing  explanations  will  make  the  following  table  of  inter- 
vals clear.     (See  page  5.) 

Intervals  may  also  be  divided  into  diatonic  and  chromatic.  Dia- 
tonic are  to  be  found  in  the  major  scale  and  in  the  natural  minor 
scale ;  chromatic  result  from  the  introduction  of  sound's  foreign  to 
the  scale. 

Questions  on  Chapter  I. 
What  is  the  smallest  interval  in  the  scale  ? 

What  is  the  difference  between  a  chromatic  and  a  diatonic  half-tone  ? 
What  is  an  enharmonic  change? 
What  is  an  interval  ? 

What  determines  the  name  of  an  interval  ? 

Give  the  names  of  the  intervals  included  within  the  limits  of  the  octave. 
What  terms  are  used  to  distinguish  between  the  different  kinds  of  intervals 

with  the  same  name? 
How  are  intervals  farther  classified? 
What  is  a  consonant  interval? 
What  is  a  dissonant  interval? 

What  farther  division  is  made  of  consonant  intervals? 
What  is  a  perfect  consonance? 
What  is  an  imperfect  consonance? 
How  is  a  diminished  interval  produced? 
How  is  an  augmented  interval  produced? 

What  is  the  nature  of  all  augmented  and  diminished  intervals? 


4 


HARMONY. 


What  is  meant  by  the  resolution  of  a  dissonance? 
In  what  kind  of  dissonances  must  one  member  resolve? 
In  what  kind  of  dissonances  must  both  members  resolve? 
When  both  members  of  the  dissonance  must  resolve,  in  what  case  do  they  ap- 
proach each  other?    In  what  case  do  they  separate? 
Are  there  five  kinds  of  every  interval? 
How  many  kinds  of  seconds  are  there? 
WThat  is  the  smallest  one  called? 
How  many  half-tones  does  it  include? 
How  many  half-tones  in  the  major  second? 
How  many  half-tones  in  the  augmented  second? 
How  many  kinds  of  thirds  are  there? 
How  many  letters  are  included  in  a  third? 
What  is  the  smallest  third  called? 
How  many  half-tones  does  it  include? 

What  is  the  difference  between  a  diminished  third  and  a  major  second? 

How  many  kinds  of  fourths  are  there? 

What  is  the  smallest  one  called? 

How  many  half-tones  does  it  include? 

How  many  half-tones  in  a  perfect  fourth? 

How  many  half-tones  in  an  augmented  fourth? 

What  does  inverting  an  interval  mean? 

What  interval  results  from  any  given  interval  and  its  inversion  taken  together? 
How  many  half-tones  are  there  in  an  octave? 
How  many  half-tones  in  an  interval  and  its  inversion? 
Given  an  interval,  how  is  the  number  of  half-tones  in  its  inversion  found? 
What  is  the  name  of  the  interval  produced  by  inverting  a  second?    A  third? 
A  fourth? 

What  kind  of  interval  is  produced  by  inverting  one  that  is  major?    One  that 
is  minor?    Perfect?    Augmented?  Diminished? 


NOTES. 

( i.)  These  questions  must  be  asked  over  and  over,  and  the  various  intervals  must  be  written, 
until  the  whole  chapter  is  understood  and  committed  to  memory. 

( 2.)  The  half-tone  is  used  as  a  measure  for  the  intervals,  it  being  more  convenient  than  it 
would  be  to  use  both  whole  and  half-tones. 

(3.)  Whole-tone  and  half-tone  are  used  in  preference  to  the  German  whole-step  and  half-step, 
their  meaning  being  perfectly  clear  ever  since  they  were  first  used  in  English. 

(4.)  Dissonant  intervals  are  by  many  writers  called  discords.  A  dissonant  is  pleasant;  a 
discord  is  something  that  never  should  appear  in  music. 

(  5.)  The  perfect  fourth  and  fifth  are  by  some  writers  called  major.  The  diminished  fifth  is 
also  called  imperfect.  The  augmented  sixth  is  called  extreme,  or  extreme  sharp  sixth.  The 
names  adopted  in  this  work  seem  more  logical  and  less  likely  to  contuse  the  student. 


HARMONY. 


5 


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6 


HARMONY. 


CHAPTER  II. 
The  Major  Scale. 

A  Scale  (from  Latin,  ^/«  =  a  ladder)  is  a  succession  of  sounds  gradually 
rising  in  pitch  from  any  given  sound  to  its  octave. 

A  Chromatic  scale  is  one  that  ascends  (or  descends)  by  half-tones. 

A  Diatonic  scale  is  one  that  ascends  (or  descends)  by  whole  and 
half-tones. 

Two  kinds  of  diatonic  scale  are.  used  in  the  modern  musical  sys- 
tem :  ist,  called  the  Major  Scale;  2d,  called  the  Minor  Scale.  1st, 
the  Major  Scale :  in  this  succession  a  half-tone  is  found  between  the 
third  and  fourth,  and  seventh  and  eighth  degrees  ;  consequently,  if  the 
scale  is  divided  into  two  groups  each  containing  four  sounds,  it  will 
be  found  that  these  two  groups  are  alike  in  consisting  of  two  succes- 
sive whole-tones  followed  by  a  half-tone,  and  that  these  two  groups 
are  separated  by  a  whole-tone,  thus : 

C,  £>,  I   G,  A,  B^C. 

These  groups  are  called  TetrachordS  (from  two  Greek  words  meaning  four 
strings) . 

A  major  scale  may  therefore  be  defined  as  being  formed  of  two 
Tetrachords  separated  by  a  whole-tone. 

As  the  Tetrachord  must  consist  of  a  succession  of  two  whole-tones 
followed  by  a  half-tone,  if  it  begins  on  any  letter  but  C  or  G  it  is 
necessary  to  alter  the  pitch  of  one  or  more  of  the  letters  to  make  it 
conform  to  this  succession.  It  is  for  this  purpose  that  sharps,  flats, 
double  sharps,  and  double  flats  are  used.  For  example,  if  it  be  desired 
to  write  a  Tetrachord  beginning  on  A,  the  succession  of  letters  will 
be  A,  B,  C,  D;  but  as  the  interval  between  B  and  C  is  a  half-tone, 
and  that  between  C  and  D  a  whole -tone,  it  is  necessary  to  move  the 

C  closer  to  the  D ;  therefore  it  is  written  A,  B,  C§  D. 

If  a  Tetrachord  is  to  be  written  beginning  on  B,  the  letters  will  be 


HARMONY. 


7 


F,  G,  A,  B ;  but  as  j5  is  a  whole-tone  above  A  its  pitch  must  be 
lowered  by  making  it  flat;  thus,  F,  G,  A,  2>b. 

By  examining  the  following  series  of  sounds  it  will  be  found  that 
the  Tetrachord  that  makes  the  second  half  of  the  scale  of  C  is  also 
the  first  half  of  the  scale  of  G,  and  the  Tetrachord  that  makes  the 
first  half  of  the  scale  of  C  is  also  the  second  half  of  the  scale  of  F. 


Scale  of  F. 


Scale  of  G. 


L_ — _ — g^Iei- 



i 


Scale  of  C. 

Therefore,  Every  Tetrachord  belongs  to  Two  Scales,  that  of  the  letter 
it  begins  with  and  that  of  the  letter  it  ends  with.  These  two  scales 
are  called  related  scales. 

Therefore  every  major  scale  has  two  related  major  scales,  one  be- 
ginning on  the  last  note  of  the  first  Tetrachord,  the  other  on  the  first 

note  of  the  second  Tetrachord.  (The  easiest  way  to  remember  is,  the  related  major 
scales  begin  on  the  fourth  and  fifth  of  the  given  scale.) 

Every  Tetrachord  must  contain  four  successive  letters,  and  every 
scale  must  contain  all  seven  letters  and  the  octave  of  the  letter  it  begins 
with. 

It  is  this  that  necessitates  the  use  of  double  sharps  and  double  flats. 
Thus,  if  a  Tetrachord  is  to  begin  on  D§,  the  letters  will  be        F,  F, 

G ;  but  to  make  a  Tetrachord  they  must  be,  F3i~ lp— j^=t^=t==:|:| 
Or,  if  the  Tetrachord  begins  on  F\>,  the  letters  will  be  F\>,  G,  A, 
B\  but  they  must  be  written. 


Questions  on  Chapter  II. 

What  is  a  scale? 

What  is  a  chromatic  scale? 

What  is  a  diatonic  scale? 

How  many  kinds  of  diatonic  scale  are  used? 

Where  do  the  half-tones  occur  in  the  major  scale? 

How  may  a  major  scale  be  divided? 


s 


HARMONY. 


Of  what  do  these  groups  consist? 
What  are  they  called? 

What  definition  may  be  given  of  a  major  scale? 

Of  what  use  are  sharps,  double  sharps,  flats,  and  double  flats? 

To  how  many  scales  does  a  Tetrachord  belong? 

What  determines  the  scales  to  which  it  belongs? 

What  are  these  scales  called? 

How  many  major  relations  has  a  given  major  scale? 
Upon  which  notes  of  the  given  scale  do  the  related  scales  begin? 
The  pupil  should  write  Tetrachords  beginning  with  every  sharp,  flat,  and  natu- 
ral, and  should  write  over  them  the  names  of  the  scales  to  which  they  belong. 


HARMONY. 


9 


CHAPTER  III. 

Major  Scale  Continued,  and  Common  Chord. 

The  following  names  are  given  to  the  degrees  of  the  diatonic  scale  : 
First,  or  keynote,  is  called  the  Tonic ; 

Second,  SllpertOniC  (upon  or  over  the  tonic)  J 

Third,  Mediant; 
Fourth,  Subdominant ; 
Fifth,  Dominant ; 
Sixth,  Submediant; 
Seventh,  Leading  Note. 

Mediant  (from  Latin  medius=  middle)  is  so  called  because  it  is  half-way 
between  the  tonic  and  the  fifth  above,  or  dominant. 

Subdominant  (Latin  sub  =  under)  is  so  called  because  it  is  the  same 
distance  below  the  Tonic  (a  fifth)  that  the  Dominant  is  above. 

Dominant  (Latin  dominans  =  ruling)  is  so  called  because  its  harmonies 

rule  or  determine  the  Scale.  (Originally  the  dominant  was  so  called  in  the  eccle- 
siastical system  because  it  was  the  principal  reciting  note  of  the  chant.) 

Submediant  is  so  called  because  it  is  half-way  between  the  tonic 
and  the  subdominant. 

Leading  Note  is  so  called  on  account  of  its  tendency  to  ascend  to 

the  keynote.  (This  note  is  also  called  the  subtonic,  in  Latin  nota  sensibihs,  in  French  note 
sensible  =  sensitive  note,  on  account  of  this  tendency.) 

COMMON  CHORD,  OR  PERFECT  CHORD  OR  TRIAD. 

A  Common  Chord  consists  of  three  sounds.  The  sound  on  which 
the  chord  is  built  is  called  the  Root  ;  the  next  is  the  Third  above  the 
root;  the  next  is  the  Fifth  above  the  root. 

The  third  over  the  root  may  be  major  or  minor  :  if  major,  the  chord 
is  called  a  Major  Chord;  if  minor,  the  chord  is  called  a  Minor  Chord. 

The  fifth  over  the  root  must  be  perfect ;  if  it  is  not,  the  chord  is  not 
a  common  or  perfect  chord. 


HARMONY. 


As  there  are  only  seven  letters  used  in  music,  it  follows  that  seven 
groups  of  letters  must  make  all  the  common  chords  possible,  as  each 
letter  may  be  the  root,  or  the  third,  or  the  fifth  of  some  chord,  thus: 

Fifths,    E,  E,     G,     A,    B,    C,  D. 

Thirds,  C,    D,    E,    E,     G,    A,  B. 

Roots,    d,    £,     C,     Z>,    E,    E,  G. 

It  will  be  seen  that  to  form  a  chord  on  any  given  letter,  for  exam- 
ple, A,  it  is  only  necessary  to  skip  one  letter  to  find  the  third,  C,  then 
skip  another  letter  to  find  the  fifth,  E ;  thus,  A  (B)  C(D)E.  There- 

^  '  s  s 

fore,  there  is  an  interval  of  a  third  between  the  root  and  third,  and 
an  interval  of  a  third  between  the  third  and  fifth.  If  from  I  to  3 
is  a  major  third,  from  3  to  5  is  a  minor  third ;  if  from  1  to  3  is  a  minor 
third,  from  3  to  5  is  a  major  third  ;  if  from  1  to  3  and  3  to  5  are  both 
minor  thirds,  the  chord  is  called  a  Diminished  Chord  (or  imperfect )  ;  if 
from  1  to  3  and  3  to  5  are  both  major  thirds,  the  chord  is  called 

an  Augmented  Chord    (  neither  are  common  chords)  . 

In  every  major  scale  Six  common  chords  may  be  written,  (i.e. with- 
out using  any  accidentals) .    Three  of  these  chords  are  major,  three  minor. 
The  chords  take  the  names  of  the  degrees  of  the  scale  upon  which 
they  are  written  ;  therefore  the  chord  on  the 
First  is  called  the  Tonic  Chord,  and  is  Major ; 
Second  is  called  Supertonic  Chord,  and  is  Minor  ; 
Third  is  called  Mediant  Chord,  and  is  Minor  ; 
Fourth  is  called  Subdominant  Chord,  and  is  Major ; 
Fifth  is  called  Dominant  Chord,  and  is  Major ; 
Sixth  is  called  Submediant  Chord,  and  is  Minor. 
The  seventh  or  leading  note  may  not  be  used  as  a  root,  because  the 
fifth  over  it  is  diminished.     (Diminished  and  augmented  chords  will 
be  treated  of  in  the  proper  place.) 

2.         3.        4.  5.         6.         7.  8. 


* 


&  g- 


N.  B. 


N.  B.  If  B  is  taken  as  a  root  the  perfect  fifth  is  E§,  which  does 
not  exist  in  the  scale  of  C. 

It  is  important  to  remember  that  no  change  in  the  order  in  which 


HARMONY. 


the  three  letters  forming  a  chord  are  written,  changes  the  name  of 
the  chord.  Thus  the  following  example  contains  the  chord  of  A 
only,  because  each  group  consists  of  the  letters  A,  C,  E. 

-<s>- 

—   Z?  g?  II 

&  _  %  IJ 

Therefore,  to  rind  the  root  of  a  chord  arrange  the  letters  composing 

it  to  read  (upwards)  One,  three,  five.  (This  rule  should  be  borne  in  mind;  it 
will  be  extended  farther  on.) 

The  pupil  should  be  required  to  write  major  and  minor  chords,  using-  every  natural,  sharp,  and 
fiat  as  roots.   The  following  remarks  will  aid  in  remembering  the  perfect  fifths. 

I.  Every  fifth  that  may  be  struck  on  two  white  keys  is  perfect, 
except  B-F;  to  make  a  perfect  fifth  between  these  letters,  either  B 
must  be  flat  or  F  sharp. 

II.  Every  fifth  that  may  be  struck  on  two  black  keys  is  perfect. 

III.  There  are  only  two  perfect  fifths  that  have  the  root  a  white 
and  the  fifth  a  black  key,  viz.,  B-F§  and  Cb-  G\> ;  only  two  that 
have  the  root  a  black  and  the  fifth  a  white  key,  viz.,  Bb-F  and  A\- 

Write  the  following  chords  in  their  natural  positions,  that  is,  root, 
third,  fifth,  and  mark  every  chord  major  or  minor,  as  the  case  may 
be. 


Questions  on  Chapter  III. 
Give  the  names  of  the  degrees  of  the  diatonic  scale. 

What  does  supertonic  mean ?    Mediant?    Subdominant?    Dominant?  Sub- 
mediant? 

Why  is  the  seventh  degree  called  the  leading  note? 
Of  how  many  scunds  does  a  common  chord  consist? 
What  is  the  sound  on  which  the  chord  is  built  called  ? 


12 


HARMONY. 


What  is  the  interval  between  the  root  and  the  next  member  of  the  chord  ?  Be- 
tween the  root  and  the  third  member  of  the  chord  ? 
What  kind  of  third  may  a  chord  have? 

What  is  the  chord  called  when  the  third  is  major  ?    What  when  the  third  is 
minor? 

What  kind  of  fifth  must  a  chord  have? 

How  many  groups  of  letters  make  all  the  common  chords? 

What  is  the  interval  between  the  third  and  fifth  of  a  chord  ? 

What  kind  of  third  is  this  in  a  major  chord  ?    In  a  minor  chord  ? 

If  both  these  thirds  are  minor  what  is  the  chord  called  ? 

If  both  are  major  what  is  the  chord  called  ? 

Are  either  of  these  common  chords? 

How  many  common  chords  may  be  written  in  a  major  scale? 

How  many  are  major  ?    How  many  minor  ? 

How  are  the  chords  named? 

Give  the  names  of  the  major  chords. 

Why  may  not  the  leading  note  be  used  as  a  root? 

Does  changing  the  order  in  which  the  letters  of  a  chord  are  written  change  its 
name? 

How  may  the  root  of  a  group  of  letters  that  form  a  chord  be  found  ? 

Note.  The  teacher  should  insist  that  when  pupils  are  giving  the  letters  that  form  a  chord, 
they  must  mention  the  sharps  or  fiats  always. 


HARMONY. 


13 


CHAPTER  IV. 
Position  and  Succession  of  Common  Chords. 
The  lowest  note  of  any  group  sounded  together  is  the  Bass. 

(This  word  should  be  spelled  base,  because  the  proper  meaning  of  bass  is  a  deep  sound.) 

It  is  not  at  all  necessary  that  the  bass  note  of  a  group  be  written 
on  the  "bass"  staff. 

Avoid  any  confusing  of  Bass  with  Root.  The  root  may  be  the 
bass,  but  the  bass  may  be  any  member  of  the  chord.  The  first  step 
in  learning  to  use  chords  is  to  learn  how  to  write  successions  in 
which  the  Roots  are  always  the  Bass  Notes. 

The  most  effective  harmony  is  that  written  in  four  parts  (or  for 
four  "voices").  Since  the  common  chord  has  only  three  letters  in 
it,  it  is  necessary  to  repeat  one  of  the  letters  to  make  a  fourth  part. 
When  the  root  is  used  as  a  bass,  the  root  itself  is  the  best  member  to 
repeat ;  thus,    C,  E,  C7,  C.     It  will  be  evident  that  while  retaining 

the  root  C  as  a  bass,  it  is  possible  to  make  three  different  arrange- 
ments of  the  remaining  letters.  Thus,  instead  of  having  the  repeated 
root  at  the  top,  the  third,  E,  might  be,  or  the  fifth,  C7,  might  be. 
Therefore  we  have  this  rule : 

Every  chord  with  its  Root  at  the  Bass,  and  with  the  Root  repeated, 
may  be  written  in  Three  Positions. 

If  the  repeated  root  is  at  the  top, 
it  is  called  the  Octave  Position  (1)  ; 
if  the  third  of  the  chord  is  at  the  top, 
it  is  called  the  Tierce  Position  (2)  ; 
if  the  fifth  is  at  the  top,  it  is  called 
the  Quint  Position  (3). 

For  the  present  these  positions  are  indicated  by  placing  over  the 
top  of  the  group,  8  for  octave,  3  for  tierce,  and  5  for  quint  position. 

(Pupils  should  be  required  to  do  this  until  they  are  quite  familiar  with  the  positions.) 


HARMONY. 


It  is  necessary  to  remember  that  the  term  Position  always  means 
that  the  Root  is  at  the  bass.  The  first  and  most  important  rule  to  be 
observed  when  writing  a  succession  of  chords  in  Positions  is  —  Two 
Chords  must  never  occur  in  Succession  in  the  Same  Position. 


< 

^    <S>  25  

a. 

 — a.  & — i_ps 

(5.                    c.  d 

 *—&-\*  1 

"7  f> 

(3;.  The  first  chord  being  in  the  octave  position,  the  following  chord 
may  be  in  either  the  third  or  fifth  position. 

b.  First  chord,  third  position;  the  next  may  be  octave  or  fifth. 

c.  First  chord,  fifth  position;  the  next  may  be  third  or  octave. 

d.  Both  chords  in  octave  position  forbidden. 

e.  Both  chords  in  third  position  forbidden. 

f.  Both  chords  in  fifth  position  forbidden. 

Consequently  there  is  always  a  choice  of  two  positions;  viz.,  either 
of  those  in  which  the  last  chord  is  not  written. 

In  general  a  better  effect  is  produced  when  the  three  upper  parts 
move  in  the  opposite  direction  to  the  bass,  especially  when  the  bass 
moves  only  one  degree.  (More  exact  rules  for  this  will  be  given 
later.) 


 ?d 

9t 


m 


EE 


§ 


to. 


I  [, 


The  motion  from  first  to  second  chord  is  called  oblique  —  that  means 
one  part  (the  top)  is  statioiiary,  the  bass  moves.  2  to  3  is  contrary 
motion;  the  bass  (a)  moves  one  degree.  3  to  4  is  direct  or  parallel 
motion  ;  bass  and  top  both  ascend.     At  b  the  bass  moves  one  degree. 


HARMONY.  15 

The  following  basses  are  to  have  the  chords  written  over  them  in 
accordance  with  the  foregoing  rules.  Do  not  be  satisfied  with  simply 
getting  the  chords  right,  but  try  the  effect  of  different  positions. 


II. 

3*B    :  g 

d — 1 

PS 

-4— i  4~ 
*  - 

— j 

1  1 

11 

L  ^  L 

III. 

EEEFEEt: 

— t 

-  d 

IV. 

-4- 

— 

2=t= 

± 

0- 

-  t 

— 1  # 

..-  •'  ! 

— F 

V. 

==l=  : 

i  

— h— r- 

1 

 \  — 

~m  0- 

II 

VI. 

==^7^=1= 

3  fj 





These  exercises  should  be  transposed  repeatedly  and  rewritten. 


Questions  on  Chapter  IV. 

What  is  a  bass  note?    Must  it  be  written  on  the  bass  staff? 

How  is  four  part  harmony  made  from  common  chords?    Which  member  is 

the  best  to  repeat  when  the  root  is  at  the  bass? 
Do  bass  note  and  root  mean  the  same  thing? 
What  is  meant  by  the  position  of  a  chord  ? 

In  how  many  positions  may  a  chord  with  the  root  repeated  be  written? 

What  are  they  called  and  why?  Which  member  of  the  chord  is  at  the  bass  in 
all  positio?is?  Give  the  first  rule  for  writing  successions  of  chords  in  posi- 
tions. 

If  a  chord  is  in  the  octave  position,  what  positions  may  the  next  chord  have? 
What  if  it  is  in  the  third  position?    What  if  it  is  in  the  quint  position? 

What  is  meant  by  oblique  motion?  By  direct  or  parallel  motion?  By  con- 
trary motion? 

Which  kind  of  motion  is  considered  the  best? 

In  what  case  is  it  especially  to  be  observed? 

Name  the  keys,  the  tonic,  subdominant,  and  dominant  chords,  and  leading 
notes  of  the  exercises  marked  I.  to  VI. 


i6 


HARMONY. 


CHAPTER  V. 


Sequences  of  Common  Chords. 


The  movement  of  common  chords  is  absolutely  free;  that  is,  any 
chord  in  the  scale  may  be  followed  by  any  other  one.  Some  of 
these  successions  will  be  found  to  be  much  smoother  than  others, 
but  there  is  no  succession  that  cannot  be  made  to  sound  well  when 
used  in  the   right  place.      For    example,   the    following  passage 


— 1:|  may  be  harmonized  in  the  following  ways  : 


1  is  smooth  but  commonplace. 

2  is  more  vigorous. 

3  is  rugged  almost  to  harshness. 

4  is  dignified,  with  a  savor  of  quaintness. 

In  2  the  progression  from  first  to  second  chord,  (i.  e.,  from  a  major 

chord  to  the  minor  chord  on  the  third  above,)  was  not  considered  good  at  one  time. 

In  3  the  progression  from  first  to  second   chord,    (i.  e.,  from  a  minor 

chord  to  another  minor  chord  one  degree  above  or  below,)  was  forbidden.  The  pro- 
gression from  the  second  to  the  third  chord,  (i.  e.,  from  a  minor  chord  to  the 
major  chord  a  third  higher,)  was  not  considered  good. 

No  rule  can  be  given  as  to  when  one  of  these  harsh  progressions 
will  sound  well  and  when  it  will  not.  The  judgment  of  the  composer 
as  to  the  effect  he  wishes  to  produce  is  the  only  guide  ;  but  pupils  should 
avoid  this  class  of  successions  until  they  have  thoroughly  learned  all 
those  that  are  smooth  and  natural. 


HARMONY. 


There  is  one  other  progression  that  should  be  avoided;  viz.,  from 
the  subdominant  to  the  dominant,  except  the  subdominant  is  in  the 
eighth  and  the  dominant  in  the  fifth  or  third  position,  thus: 

5  3 

N.  B.  The  above  remarks  must  be 
understood  as  applying  only  to  Suc- 
cessions of  chords  in  Positions  ;  i.  e., 
with  their  roots  at  the  bass. 

Common  chords  are  frequently  written  in  what  are  called  Sequen- 
ces (Latin sequens=  following).    In  a  sequence  the  roots  move  in  a  regular 


order  or  pattern,  for  example  : 


-a — 


P 


up  four,  down  three.  The  result  is  three  pairs  of  chords,  each  one 
degree  higher  than  the  last. 

The  most  usual  sequences  are  here  given,  both  ascending  and  de- 
scending. The  pupil  should  write  them  in  various  keys  and  should 
also  exercise  his  ingenuity  in  constructing  others. 


a,  6,  c,  are  ascending  sequences. 
d,  e,  f,  are  descending  sequences. 


i.S 


HARMONY. 


It  is  allowable  to  use  the  dimin- 
ished (leading  note)  chord  in  a 
sequence  for  the  sake  of  preserving 
the  "pattern,"  thus: 

The  use  of  the  diminished  chord 
was  sometimes  "dodged"  by  the 
older  writers  in  the  following  way  : 

The  jE>Q  is  retained  in  the  chord 
of  G  because  B\  would  necessitate 
a  chromatic  progression,  B\>,  2?tj, 
which  was  forbidden  in  strict  coun- 
terpoint. 

Sequences  composed  of  common  chords  natural  to  the  scale  are 
called  Diatonic  or  Contrapuntal.  They  were  much  more  largely 
used  by  the  old  school  than  by  the  new  ;  still  it  is  absolutely  necessary 
that  the  pupil  should  familiarize  himself  with  them.  The  reason  why 
will  appear  farther  on. 

The  Positions  of  the  chords  may  be  changed  in  all  these  examples; 
thus,  Example  a,  page  17,  may  be  jive,  three,  or  three,  eight,  or 
three,  Jive,  or  eight,  three,  alternately,  thus: 

5353         3838         3535         8     38  3 

This  is  not  as 
good  as  the  others. 


Observe  that  the  best  effect  is  produced  when  the  outer  parts  move 
in  opposite  directions. 

It  does  not  sound  well  to  move  to  a  chord  in  the  Octave  Position 
with  the  Bass  and  upper  part  (soprano)  moving  in  the  same  direction, 
except  in  the  following  cases: 

I.  From  Dominant  to  Tonic  Chord. 

II.  From  Tonic  to  Subdominant  chord.  (The  relation  between  these  chords 
is  the  same  as  that  of  the  previous  chords.) 


HARMONY. 


l9 


 g2. 

— r 


i 


Another  rule  is  often  given,  viz.,  that  the  eighth  may  occur  in 
Direct  Motion  when  the  upper  part  ascends  a  half-tone  ;  but  whenever 
this  is  the  case,  it  must  be  either  a  progression  from  Dominant  third 
position  to  Tonic  octave,  or  Tonic  third  position  to  Subdominant  octave. 

N.  B.  It  is  owing  to  the  infraction  of  this  rule  that  this  passage  does 
not  sound  well.  Another  progression  that  does  not  sound  well  is 
that  to  a  Chord  in  the  Quint  Position  with  the  outer  parts  (bass  and 
soprano)  moving  in  the  same  direction,  except  in  the  following  case: 

From  any  chord  to  another,  the  roots  of  which  chords  are  a  fifth 
apart  (ascending),  or  what  is  the  same,  a  fourth  apart  (descending). 


4= 


r 

i 


__L 

--r 


i 


Tonic.  Dominant. 


Subdom-  Tonic, 
inant. 


:=J: 


Two  chords  may  be  WTitten  in  succession  in  the  Tierce  Position  by 
doubling  the  third  of  one  of  them. 

If  the  roots  ascend,  double  the  third  of  the  Second  chord. 
If  the  roots  descend,  double  the  third  of  the  First  chord. 

V         ^  -<5»J  .&r         -&r\       -©J         °         &  -&r 

(What  sequences  may  be  made  of  chords  in  the  third  position?) 


20 


HARMONY. 


Two  chords  may  be  written  in  succession  in  octaves,  fifths,  or  thirds 
when  the  outer  parts  move  in  opposite  directions. 

8       8         5        5         3  3 

1         The  last  example  (3)  is  fre- 


9± 


2. 


quently  used.     2   is  the  most 
unusual.     1  is  common  at  the 
|_|    end  of  a  piece. 


Exercises. 

The  pupil  should  be  required  to  point  out  all  the  sequences  that  may  be 
found  in  these  exercises,  also  the  places  where  examples  of  all  the  progres- 
sions mentioned  in  this  chapter  may  be  used.  For  example,  in  the  first  exercise 
the  first  two  measures  form  a  sequence  ;  the  three  chords  marked  x  may  all  be 
in  the  third  position. 

I. 


— I- 


Avoid  unnecessary  skips  from  one  chord  to  the  next 
II. 


iH§8 


III. 


^2: 


3 


IV 


si 


V. 


E=Id2:2: 
VI. 


2^: 


zest 


2=t=±E 


11 


HA  RMONY. 


21 


VII. 


1  1  Li.  i_2  § 

m — -0  \—S  

"n    — ^ 

1  1—  M  

— 1  «  <p — |  1  1  

r  *  ^ -=3  li 

+     £    *    t»  +1 

VIII. 


1  

 £2- 

It 

-0  T 

h- 

-TP- 

-  II 

Questions  on  Chapter  V. 

Is  there  any  restriction  on  the  movement  of  common  chords? 
Give  the  successions  that  are  mentioned  as  lacking  in  smoothness. 
What  is  a  sequence  ? 

Is  the  use  of  the  diminished  chord  permitted  in  a  sequence? 

What  are  sequences  composed  of  chords  natural  to  the  scale  called  ? 

Give  the  cases  in  which  an  octave  position  may  be  taken  in  direct  motion  be- 
tween the  outer  parts. 

Give  the  cases  in  which  a  fifth  position  may  be  taken  in  direct  motion  between 
the  outer  parts. 

How  may  two  chords  be  written  in  succession  in  the  third  position? 
Which  of  the  chords  has  the  third  doubled  when  the  roots  ascend  ? 
Which  when  the  roots  descend  ? 

State  the  cases  when  two  chords  may  be  written  in  succession  in  octave,  fifth, 
and  third. 

Note.    When  writing  exercises  draw  a  slur  >  under  all  the  roots  that  form  the  sequence^ 

also  under  succession  of  chords  in  third  position. 


3  2 


HARMONY. 


CHAPTER  VI. 


First  Inversion  of  Common  Chords. 


If  the  preceding  chapters  have  been  thoroughly  mastered,  the  pupil 
will  know  all  that  may  be  done  with  the  Common  Chords  natural  to 
the  Scale,  with  their  Roots  as  Bass  notes;  i.  e.,  in  Positions. 

The  next  step  is  to  Invert  the  chords.  Inversion  means  using  some 
other  member  of  the  chord  as  a  bass. 

If  the  Third  of  the  chord  is  used  as  a  bass,  the  Chord  is  said  to  be 
in  its  First  Inversion. 

The  First  Inversion  of  a  Chord  is  subject  to  no  restrictions;  that 
is,  any  chord  in  the  scale  may  be  written  with  its  third  at  the  bass. 

When  the  Third  is  used  as  a  bass,  the  Root  or  Fifth  may  be  re- 
peated (to  make  the  fourth  part) ;  it  is  immaterial  which. 


b. 


d. 


a.  Chord  C,  E,  G,  with  the  third, 
E,  at  the  bass,  the  root  repeated. 

b.  Same,  with  the  root  repeated 
at  unison.  This  is  signified  by 
doubling  the  note. 

c.  Same,  with  the  fifth  repeated. 
•  d.  Same,  with  fifth  repeated -at  unison. 

It  is  a  rule  that  No  two  Voices  or  Parts  must  ever  move  together 

in  Fifths  Or  Octaves  (making  what  are  called  parallel  or  consecutive  fifths  or  octaves). 
(This  rule  will  be  largely  modified  in  the  proper  place,  but  must  be  strictly  observed  at  present. ) 

It  is  in  consequence  of  this  rule  that  it  is  forbidden  to  write  two 
chords  in  succession  in  the  same  position. 

a.  Both  in  octave  position; 
therefore  the  Jirst  and  fourth 
parts  move  together  in  octaves, 
the  second  and  fourth  in  fifths. 

b.  Both  in  third  position; 
second  and  fourth  in  octaves, 
third  and  fourth  in  fifths. 


HARMONY. 


c.  Both  in  fifth  position,  first  and  fourth  in  fifths,  third  and 
fourth  in  octaves. 

The  doubling  of  the  third,  when  two  chords  are  written  in  suc- 
cession in  the  tierce  position,  is  sure  to  avoid  the  parallel  fifths  and 
octaves. 

In  writing  the  same  Position  twice  in  succession,  the  parallel  fifths 
or  octaves  will  always  be  found  between  the  bass  and  one  of  the 
upper  parts.  If  two  First  Inversions  are  written  in  succession,  the 
parallel  fifths  and  octaves  will  occur  between  two  of  the  upper  parts, 
thus  : 

a.  Octaves  between  first  and 
third  parts,  fifths  between  second 
and  third,  caused  by  repeating  the 
root  in  both  chords. 

b.  Octaves  between  first  and 
third  parts,  fifths  between  first  and 
second,  caused  by  repeating  the 
fifth  in  both  chords. 

Therefore  they  may  be  avoided  by  repeating  the  Root  in  one 
chord  and  the  fifth  in  the  other,  or  repeating  one  at  unison,  the 
other  at  the  octave. 


a.  Both  roots  repeated —  one  at  unison,  the  other  at  octave. 

b.  Fifth  at  unison,  root  at  octave. 

c.  Root  at  octave,  fifth  at  octave ;  but  the  repetition  of  the  root 
makes  an  octave  between  the  frst  and  third  parts,  that  of  the  fifth 
an  octave  between  the  second  and  third  parts,  and  to  be  parallel, 
they  must  occur  between  the  same  two  parts. 

d.  Both  fifths  repeated.  The  first  and  second  parts  being  in 
unison,  there  is  a  fifth  between  them  and  the  third  part;  but  the 
second  fifth  is  between  the  first  and  second parts.          -  .  .. 


HARMONY. 


e.  The  repeated  C  is  root  of  one  and  fifth  of  the  other  chord ;  but 
these  octaves  are  not  consecutive  because  they  are  stationary.    To  be 
consecutive  they  must  move. 
f  and g.  Roots  repeated,  unison  and  octave. 

Parallel  fifths  and  octaves  may  also  be  avoided  by  repeating  the 
Third  of  one  or  both  of  the  inverted  chords,  provided  the  repetition 
of  the  Third  is  made  by  two  parts  moving  in  opposite  directions. 

The  Third  of  a  chord  may  be  repeated  at  any  time,  if  the  repeti- 
tion occurs  in  parts  moving  in  opposite  directions. 


s 


j- 


3  I 


1 


ipzzzzp 


a.  First  and  second  chord 
inverted,  the  third  of  2  re- 
peated. 

b.  Second  chord  only  invert- 
ed, its  third  repeated. 

c.  Third  of  second  chord  repeated. 

d.  Same. 

e.  The  third  of  the  same  chord  repeated.  This  case  is  peculiar  as 
it  really  makes  parallel  octaves  between  the  first  and  fourth  parts,  but 
they  occur  in  opposite  (  contrary  )  motion.  The  long  skip  of  the 
bass  also  helps  to  hide  them. 

f.  The  third  repeated  in  both  chords. 

The  above  examples  should  be  written  in  all  the  keys,  until  they 
are  thoroughly  understood  and  memorized.  Then  the  exercises  fol- 
lowing should  be  written,  care  being  taken  to  introduce  examples  of 
all  the  progressions  here  given. 

Any  bass  note  may  be  treated  as  a  Root,  or  as  a  Third.*  When  it  is 
treated  as  a  third,  put  a  3  under  it,  and  no  figure  over  the  upper  part. 


*  Note.  It  is  perhaps  necessary  to  repeat  that  the  leading  note  cannot  be  a  root;  therefoie 
the  second  of  the  scale  cannot  be  a  third. 


HARMONY. 


25 


When  a  chord  is  Inverted  it  is  not  in  any  Position  ;  therefore  the 
chord  that  follows  it  may  be  in  any  position. 

Analyze  the  following  example,  give  the  Name,  Position,  or  Inver- 
sion of  every  chord.  State  which  member  is  repeated,  and  give  the 
reason  of  every  repetition. 


4-1 


-0-  -m- 


1— rn~r 


^-=4— •-*-r-a-#-=t 


IT 

T  ' 

A-r&  -g-- 

 0  1- 

»-F^ — #  

This  example  should  be  copied,  and  the  chords  numbered,  and  the 
analysis  of  each  chord  written  out,  thus : 

1.  Tonic  chord,  tierce  position. 

2.  Supertonic,  first  inversion,  third  doubled  by  parts  moving  in 
opposite  directions,  and  so  or. 

I. 

Fc^-^-^-^-F^-— ^-F-h — i-^P-=l=iF;^ 


II. 


III. 


IV. 

/j/ir?TifT?i  ii  iii  ! j  ;i   i  ii 


26 


HARMONY. 


V. 


VI. 

±_f:  : 

4- 

 1  1  ^ 

it  Fl 

ff  4  *  *  p- 

±=p 

 *-     *  ^  - 

Questions  on  Chapter  VI. 
What  does  the  inversion  of  a  chord  mean? 

Which  member  of  the  chord  is  at  the  bass  in  the  first  inversion? 

Which  chords  in  the  scale  may  be  used  in  the  first  inversion  ? 

Which  members  of  the  chord  may  be  repeated  ? 

What  is  meant  by  parallel  or  consecutive  fifths  or  octaves? 

Between  which  parts  will  the  parallel  fifths  or  octaves  be  found,  when  two 

chords  are  written  in  the  same  position? 
When  two  first  inversions  are  written  in  succession,  where  will  the  parallel 

fifths  or  octaves  be  found  ? 
How  may  these  be  avoided? 

What  is  the  rule  in  regard  to  the  repetition  of  the  third  ? 

Which  degree  of  the  scale  may  not  be  a  root?    Which  not  a  third? 


HARMONY. 


.27 


CHAPTER  VII. 

Second  Inversion  of  Common  Chords. 

When  a  chord  is  written  with  its  Fifth  as  a  bass  note,  it  is  called 
the  Second  Inversion  of  the  Chord. 

The  use  of  Second  Inversions  is  very  much  restricted. 

A  Chord  in  its  Second  Inversion  is  either  a  Tonic,  a  Subdominant, 
or  a  Dominant. 

The  Second  Inversion  of  a  Tonic  may  be  used  at  any  time,  pro- 
vided it  is  followed  by  the  Dominant  Chord.  (Note  the  author  is  well  aware 
of  the  rule  which  says,  it  must  never  enter  with  a  leap  in  the  bass,  but  fails  to  see  the  utility  of 
rules  that  the  greatest  composers  disregard.) 

In  Second  Inversions,  the  fifth  (bass  note)  is  the  best  to  re- 
peat. 


a.  Second  inversion  of  tonic,  preceded  by  tonic,  b.  By  supertonic. 
c.  By  mediant,  d.  By  subdominant.  e.  By  submediant.  f.  By  first 
inversion  of  supertonic  with  third  doubled.  This  succession  and  that 
at  d  are  very  smooth  and  orthodox. 

Cadence  (Latin  cado=  to  fall).  This  term  is  applied' to  various  kinds  of 
endings.  The  Perfect  or  Authentic  Cadence  is  the  final  tonic  pre- 
ceded by  the  dominant  with  its  root  at  the  bass.  This  is  emphasized 
if  the  dominant  is  preceded  by  the  second  inversion  of  the  tonic,  and 
still  more  emphasized  if  the  second  inversion  of  the  tonic  is  preceded 
by  the  subdominant  (as  at  d)  or  supertonic  (as  at  /) . 


28 


HARMONY. 


Examples  of  Perfect  Cadence. 


 2  

 ^ — 

i  1  

-<5«  

The  Second  Inversion  of  a  Subdominant  must  be  preceded  by  the 
Tonic  Chord  with  its  root  at  the  bass.  It  is  generally  followed  also  by 
the  Tonic  Chord  with  its  root  at  the  bass. 


-g 

j — l-Jf— g — 

 &  

tr     |  r 

==1  =1 

— i 

X 

X 

X 

The  Second  Inversion  of  the  Dominant  chord  is  preceded  by  the 
Tonic  Chord  with  its  root  at  the  bass,  and  followed  by  the  Tonic  Chord 
with  its  third  at  the  bass,  or  just  the  reverse.  This  second  inversion 
is  not  often  used  on  the  accent  of  the  measure.  It  is  apt,  in  this  case, 
to  sound  like  a  Tonic  chord. 


x  x  x  x 

a  and  b  are  better  than  c  and  d,  on  account  of  the  contrary  motion 
between  the  outer  parts. 

Although  the  second  inversions  always  enter  as  above,  viz.,  if  a 
tonic,  after  any  chord  in  the  scale;  if  a  subdominant,  after  the  tonic, 
root  at  bass ;  if  a  dominant,  after  the  tonic,  root  or  third  at  bass  — 
they  do  not  always  conform  to  these  examples  as  to  the  chord  that  fol- 
lows them ;  but  the  rest  of  this  subject  must  be  reserved  for  a  later 
chapter. 


HARMONY. 


29 


The  exercises  that  follow  are  to  be  transposed  to  all  the  keys. 
They  are  the  last  basses  that  will  be  given. 

I. 


Questions  on  Chapter  VII. 

What  is  meant  by  the  second  inversion  of  a  chord? 
Which  chords  may  be  used  in  the  second  inversion? 
What  must  precede  the  second  inversion  of  a  tonic? 
What  must  follow  it? 

What  must  precede  and  follow  the  second  inversion  of  a  subdominant? 
What  must  precede  and  follow  the  second  inversion  of  a  dominant? 
What  is  meant  by  a  perfect  cadence? 


3° 


HARMONY. 


CHAPTER  VIII. 

Harmonizing  of  Melodies,  with  Common  Chords  and 
Their  Inversions. 

The  knowledge  of  Common  Chords  now  gained,  must  now  be  ap- 
plied to  the  harmonizing  of  Melodies ;  first,  with  the  Chords  in  Posi- 
tions. 

As  every  chord  may  be  written  in  three  positions,  it  follows  that 
every  note  in  a  melody  may  be  either  the  root,  third,  or  fifth  of  some 
chord  belonging  to  the  scale.  If  it  is  treated  as  a  root,  the  bass  will 
be  the  same  letter ;  if  as  a  third,  the  bass  will  be  the  third  letter  be- 
low ;  if  as  a  fifth,  the  bass  will  be  the  fifth  letter  below. 

It  is  evident  that  if  two  notes  in  succession  in  the  melody  are 
treated  as  roots,  or  thirds,  or  fifths,  the  result  will  be  two  chords  in 
the  same  Position. 

Begin  and  end  with  the  Tonic  Chord.  Observe  that  the  melody 
may  begin  on  the  root,  third,  or  fifth  of  the  tonic.  Be  careful  to 
avoid  using  the  Leading  note  as  a  root. 

The  first  time  the  exercises  are  written,  observe  carefully  the  rule, 
not  to  write  two  successive  Positions  alike. 

The  second  time  they  are  written,  find  as  many  opportunities  as 
possible  to  put  two  or  three  successive  chords  in  the  Tierce  Position. 

Sequences  are  indicated  in  the  Melody  as  they  are  in  the  bass,  by 
the  notes  moving  in  a  regular  pattern.     Thus,  the  following  notes 

hEEjzzfj    indicate  the  following  Sequence: 


Others  may  readily  be  found 
by  referring  to  the  sequences  in 
Chapter  V. 


HARMONY. 


3^ 


^33 


ii. 


fcd: 


III. 


<f. 


as; 


IV. 


V. 


t=i= 


VI. 


VII 


F  F-M-p — 


VIII. 


1 


After  harmonizing  these  exercises  with  chords  in  Positions,  write 
them  over  again,  introducing  First  Inversions.  Then,  for  the  last 
time  introduce  both  First  and  Second  Inversions.  Analyze  the  fol- 
lowing example  of  a  melody,  harmonized  three  times. 


HARMONY. 


First  time,  chords  in  position. 

Second  time,  first  inversions  introduced. 

Third  time,  first  and  second  inversions  introduced. 

— i — i — — 


9t 


Play  these  over,  and  observe  how  the  musical  effect  improves  with 
the  use  of  the  inversions. 


Questions  on  Chapter  VIII. 

What  member  of  a  chord  may  any  note  in  the  melody  be? 

If  treated  as  a  root,  what  will  the  bass  note  be? 

If  as  a  third,  how  far  below  will  the  root  be  found  ?    If  as  a  fifth  ? 

May  two  notes  in  succession  be  roots,  or  thirds,  or  fifths? 

When  may  two  or  more  notes  be  thirds  ? 

What  chord  must  begin  and  end  the  piece  ? 

Note.    A  piece  does  not  always  begin  with  the  tonic  choid. 

Must  the  first  and  final  chord  be  in  the  octave  position? 


HARMONY 


33 


CHAPTER  IX. 

The  Minor  Scale. 

The  Minor  scale  historically  seems  to  have  preceded  the  Major. 
Although  its  formation  from  two  Tetrachords  is  not  so  evident  as  in 
the  Major  scale,  yet  it  is  nevertheless  true.  In  the  oldest  form  of 
Tetrachord  the  half-tone  lay  between  the  first  and  second  letters  ;  thus, 

E,   F,    G,   A.      A  seven  note  scale    (which  preceded  the  octave  scale)  was 

formed  by  joining  two  of  these  Tetrachords ;  but  the  second  Tetra- 
chord began  with  the  letter  upon  which  the  first  Tetrachord  ended ; 

thus,  E,  F,  G,  A. 

'  A,  m,  C,  D. 


This  was  called  the  scale  of  Conjunct  (joined)  Tetrachords.  It  was 
made  into  an  octave  scale  by  adding  a  sound  below  the  first  Tetra- 
chord;  thus,  D,  E,  F,  G,  A. 

A,  ,Z?b,  C,  £>.    This  was  .  known  as  the 

Dorian  scale. 

It  will  be  seen  that  the  half-tones  lie  between  the  second  and  third, 
and  fifth  and  sixth.  This  form  is  now  called  the  Natural  Minor 
Scale. 

It  will  be  found  that  by  taking  the  sounds  of  any  Major  scale  and 
arranging  them  in  arvoctave  succession  beginning  with  the  sixth  of 
the  Major  scale,  it  will  give  a  succession  identical  with  this  ;  therefore, 

Every  Minor  scale  is  called  the  Relative  Minor  of  the  major 
scale  from  which  it  is  formed,  and  it  has  the  same  signature.     (It  is 

perhaps  easier  to  remember  the  relative  minor  as  beginning  on  the  third  below  the  tonic,  this  being 
the  inversion  of  the  sixth  above.  J 

The  requirements  of  harmony  have  made  several  modifications  in 
this  scale. 

I.  It  is  necessary  to  modern  ears  to  have  a  half-tone  between  the 
seventh  and  eighth  of  a  scale  in  ascending.    If  the  seventh  is  raised, 


3.4 


HARMONY. 


there  results  the  awkward  interval  of  an  augmented  second  between 
6  and  7 ;  to  remedy  this,  6  must  also  be  raised.  But  the  need  of  a 
half-tone  between  7  and  8  is  not  felt  in  a  descending  scale ;  they  are 
therefore  allowed  to  remain  unaltered.  This  form  of  ascending  and 
descending  scale  is  called  the  Melodic,  because  the  raised  sixth  and 
natural  seventh  are  not  found  in  any  of  the  chords  belonging  to  the 
scale.      (See  note.) 

We  have  nothing  to  do  at  present  with  either  of  these  forms,  but 
with  a  third  form  called  the  Harmonic.  In  this  the  seventh  is  always 
raised,  both  ascending  and  descending.  So  the  minor  scale  is  always 
to  be  thought  of  at  present  as  having  a  raised  seventh  or  leading  note. 
These  three  forms  of  minor  scale  are  here  given. 

Scale  of  A-Minor.    The  Relative  Minor  of  C. 


Natural. 


■  0 

— -s>— 

9  bj- 

— <s»— 

-n  \ 

— rS>— 

— <S>— 

Four  common  chords  may  be  written  in  the  Harmonic  Minor  scale. 
Two  are  minor,  viz.,  Tonic  and  Subdominant;  two  major,  viz.,  the 
Dominant  and  Chord  on  sixth. 

Dominant  chords  must  be  major.  This  is  the  harmonic  reason  for 
the  raising  of  the  seventh;  hence  this  scale  is  called  the  Harmonic. 

Note.  In  the  older  writers  the  raised  sixth  was  harmonized  as  the  fifth  or  third  of  a  chord.  It 
is  rarely  found  in  modern  music.  Another  form  of  minor  scale  was  also  used  by  the  older  writers, 
called  the  mixed  minor.  In  this  scale  the  6  and  7  were  raised  both  ascending  and  descending. 
Consequently  it  differs  from  the  major  scale  only  in  having  the  third  above  the  tonic  minor.  This 
scale  will  be  found  frequently  in  Bach  and  Handel. 


HARMONY. 


3,5. 


Scale  of  A-MinoR,  Harmonic,  with  Chords, 
i.         2.         3.         4.         5.  6. 


-1- 


Tonic,  minor. 

The  fifth  is  diminished. 

The  fifth  is  augmented,  owing  to  the  raising  of  the  seventh. 
Subdominant,  minor. 


i 


1 . 

2. 

3- 
4- 
5- 


Dominant,  major.     Observe  that  this  is  the  only  chord  in  the 


scale  in  which  the  raised  note  is  found,  and  that  it  is  the  third  in  this 
chord. 

6.  Submediant,  major. 

7.  Leading  note,  diminished  fifth. 
Analyze  the  following  example : 


§  gzz ^ 


J-L-4 


4 


-<5t- 


9i 


-F- 


:t 


^  2::?:  :f:  ^  <^ 


1 


=t±E 


1 

A  Major  Scale  and  its  Relative  Minor  are  so  closely  bound  together 
that  they  may  conveniently  be  looked  upon  as  one  and  the  same  thing. 
Consequently  if  the  seventh  of  the  minor  scale  appears  not  raised,  it 
is  generally  as  a  member  of  one  of  the  chords  in  the  related  major 
scale. 


£f~®- 


3=4 


3= 


2  is  in  the  relative  major. 

3  and  4  in  minor. 

Part  of  5  and  6  in  major,  the  remainder  in  minor. 


HARMONY. 


It  was  customary  at  one  time  to  end  all  compositions  in  the  minor 
key  with  a  major  tonic.  This  major  third  was  called  the  "Tierce  de 
Picardie."    It  is  still  occasionally  used. 

In  harmonizing  the  exercises  that  follow,  find  opportunities  for  the 
use  of  the  relative  major  chords.  The  natural  seventh  of  the  minor 
scale  must  (at  present)  be  either  root  or  fifth  of  a  chord  in  the  relative 
major. 

It  is  necessary  now  to  speak  of  the  Diminished  Chord,  but  more 
will  be  said  about  it  later. 

Diminished  Chords  may  be  used  freely,  with  the  proviso  that  they 
are  used  in  the  first  inversion. 

In  the  Major  scale  there  is  but  one  ;  viz.,  on  the  Leading  note.  In 
the  Minor  scale  in  addition  to  one  on  the  Leading  note  there  is  one 
on  the  second  (Supertonic)  of  the  scale.  This  last  mentioned  chord  is 
very  important  in  the  minor  scale. 


The  Bass  note  of  this  chord  is  the 
best  to  repeat. 


g  pr- 

m 

w  § 

-! 

t 

First  inversion,  bass 
repeated. 

Diminished  chord  on  second  of  A  minor,  (it  will  be  seen  that  it  is  also  the 
leading  note  chord  of  the  relative  major.)  The  second  inversion  of  the  tonic 
enters  very  effectively  after  this  chord. 

t  Diminished  chord  on  second, 

**  first  inversion. 

X 


III. 


'i 

-<s>- 

r-Q~b—  It 

1 — i  %  

=1    ^  -i  1 

Questions  on  Chapter  IX. 

Where  do  the  half-tones  lie  in  the  natural  minor  scale? 

Why  is  a  minor  scale  called  the  relative  minor  of  a  major  scale? 

Upon  which  degree  of  the  major  does  the  relative  minor  begin? 

To  what  form  of  minor  scale  is  the  term  melodic  applied  ? 

To  what  form  is  the  term  harmonic  applied  ? 

Where  do  the  half-tones  lie  in  the  harmonic  scale? 

What  is  the  interval  between  the  sixth  and  seventh  ? 

How  many  common  chords  are  found  in  the  harmonic  scale? 

Which  are  major?  Which  minor? 

What  kind  of  fifth  does  the  mediant  bear? 

In  which  chord  is  the  raised  seventh  found? 

What  member  of  this  chord  is  it? 

If  the  seventh  is  not  raised,  how  is  it  generally  treated? 
Where  are  diminished  chords  found? 
In  what  form  may  they  be  used  ? 
Which  member  should  be  repeated  ? 


3* 


HARMONY. 


'   '  CHAPTER  X. 

The  Group  or  Circle  of  Related  Keys. 

In  Chapter  II.  the  relationship  of  Major  scales  through  the  Tetra- 
chords  is  explained.  In  the  last  chapter  the  relation  between  a 
Major  and  its  Relative  minor  is  explained.  This  relationship  must 
now  be  extended  to  include  the  Relative  Minors  of  the  Major 
Relations. 

Thus,  given  the  key  of  C,  the  related  majors  are  E,  G. 
The  relative  minors  A,  D,  E. 

Thus  a  given  key  always  includes  a  group  of  six  keys,  three  Major 
keys  and  their  Relative  Minors. 

Each  of  these  scales  must  have  a  leading  note ;  thus,  C  being  the 
key,  E,  its  first  major  relative  has  E  as  leading  note.  G,  the  sec- 
ond major  relative,  has  E§  as  leading  note;  A  minor  has  G§;  D 
minor,  Cjf ;  E  minor,  Z$. 

So  four  accidentals  may  be  introduced  in  the  scale  of  C,  as  lead- 
ing notes  to  the  related  scales.  We  found  that  the  leading  note  of 
the  minor  scale  was  only  to  be  found  in  the  dominant  chord,  in 
which  chord  it  is  the  third.  From  these  facts  we  deduce  the  follow- 
ing rule : 

Notes  raised  by  accidentals  are  Leading  notes  to  Related  Keys, 
and  they  are  always  harmonized  as  Third  in  the  Dominant  Chord  of 
the  key  to  which  they  lead. 

The  most  natural  progression  of  a  Dominant  chord  is  to  its  Tonic. 


i. 

2. 

'  3- 

4- 

5.  6. 

%  * 

— 1— 

=1  & 

A  M 



m= 

sr 

— 1  

 &  1  

& — 
_j  

F  CPHI 

I.  Dominant  and  tonic  of  D  minor,  relative  minor  of  E. 


HARMONY. 


39 


2.  Dominant  and  tonic  of  E  minor,  relative  minor  of  G.    (When  there 

are  two  raised  notes  in  a  chord,  the  last  one  to  occur  is  the  third ;  therefore  D$,  which  does  not  oc- 
cur until  after  Ft,  is  the  third,  F  must  be  sharped  in  F  minor  because  it  is  sharped  in  G  major.) 

3.  Dominant  and  tonic,  F  major. 

4.  Dominant  and  tonic,  G  major. 

5.  Dominant  and  tonic,  A  minor,  relative  minor  of  C. 

6.  Dominant  and  tonic  of  Key. 

It  is  much  to  be  regretted  that  one  word,  modulation,  is  used  to 
signify  going  into  a  related  key  transiently  (especially  by  means  of  its  domi- 
nant chord),  and  going  outside  of  the  related  groups. 

The  writer  has  thought  that  for  clearness'  sake,  the  word  modu- 
lation might  be  used  only  in  the  first  sense,  and  that  the  word 
transition  better  describes  this  passing  to  a  new  group  of  relations. 
Therefore  in  this  work  these  words  will  be  used  with  this  distinction 
between  them. 

In  all  the  modulations  in  the  foregoing  example,  the  idea  of  the 
original  key  is  never  lost  for  a  moment, — what  is  called  the  "Tonality." 
The  reason,  perhaps,  is  that  the  Tonic  chords  are  simply  the  six 
common  chords  of  the  original  scale.  If  one  of  these  Tonics,  when 
preceded  by  its  dominant  is  changed,  for  example,  if  the  tonic  of 
D  minor  (  1  )  is  changed  to  major,  the  tonality  of  the  original  key 
is  at  once  lost,  and  a  transition  is  made. 

The  following  examples  will  show  how  these  dominant  chords  may 
be  used. 


-~ !  


9* 


m 


-0r 


a.  Is  harmonized  with  chords  natural  to  the  scale. 

b.  Same  melody,  with  dominant  chords  of  related  scales  introduced. 
Harmonize  the  following  passages  in  the  same  manner ;  follow 

every  dominant  by  its  own  tonic  for  the  present. 
1.  2.  3. 


EH 


4o  HARMONY. 

i^iipi^BIsilllligi^a 

In  i  the  dominant  and  tonic  of  D  minor  may  be  used. 
In  2  the  dominant  and  tonic  of  A  minor  and  G  major. 
In  3  the  dominant  and  tonic  of  A  minor,  G  major,  JF  major. 
In  4  the  dominant  and  tonic  of  G  major,  twice;  or  second  place 
may  have  dominant  E  minor. 

In  5  the  dominant  and  tonic  of  E,  G,  A. 
In  6  the  dominant  and  tonic  of  D,  E. 

When  these  dominant  chords  of  related  keys  are  used  in  sequences, 
the  sequence  is  called  harmonic  (  or  chromatic  ) .  Sequences  will  be  found 
in  2,  J,  5,  and 6  of  the  above  examples.  A  few  more  are  here  given. 
They  and  the  foregoing  exercises  should  be  transposed  to  every  major 
key. 


Exercises  Introducing  Related  Keys. 


HARMONY. 


41 


III. 


IV. 


t=*=n 
3=d 


v. 


VI. 


fa 


±=d: 


let 


The  rules  about  second  inversions  may  now  be  extended  so  as  to  in- 
clude the  Tonic,  Dominant,  and  Subdominant  chords  of  the  Related 
keys. 


r 


X 


0 


zt: 


:pz=: 


-251- 


-(2- 


1  1  5  s- 

L-t--5-^- 

-4- 

rr  rr-r  i~ 

9z  a 

1.  Second  inversion  of  some  related  tonics. 

2.  Second  inversion  of  subdominants. 

3.  Second  inversion  of  do-minants. 

Although  the  dominant  chords  generally  progress  to  their  tonics, 


HARMONY. 

being  Common  chords  —  have  othe-r  progressions,  which 

1 


42 

they  may  - 

may  occur  singly  or  be  arranged  in  sequences, 
r.  2. 


4- 


6. 


1.  Dominant,  followed  by  the  chord  on  the  sixth  of  the  scale  to 
which  it  belongs. 

2.  Dominant  of  minor,  followed  by  tonic  of  relative  major. 
a.  Both  roots  at  bass.  b.  Tonic,  with  third  at  bass.  c.  Dominant, 
third  at  bass ;  tonic,  fifth  at  bass. 

3.  Two  dominants  in  succession.  In  this  case  the  roots  must  be  a 
fourth  apart  (ascending),  or  fifth  apart  (descending). 

4.  Dominant  of  a  minor,  followed  by  chord  on  minor  third  above. 

5.  Dominant  changed  chromatically  to  natural  chord  of  scale. 

6.  Same  as  4,  but  second  chord  with  fifth  at  bass. 

7.  Dominant  of  a  major  key,  followed  by  dominant  of  its  relative 
minor. 

1,  3,  and  7  are  of  special  importance. 
Transpose  these  examples  to  all  keys. 


Questions  on  Chapter  X. 

How  many  scales  are  included  in  the  related  group? 
How  many  are  major  ?    How  many  minor  ? 

How  many  accidentally  raised  notes  may  be  introduced  in  a  given  scale? 
What  are  these  accidentals? 

What  is  the  natural  progression  of  a  dominant  chord  ? 

What  is  the  difference  between  a  diatonic  and  a  harmonic  sequence? 


HARMONY. 


43 


CHAPTER  XI. 

Chords  of   Parallel    Minor,   Lowered   Supertonic,  and 
Chords  in  the  Related  Keys  not  Found 
in  the  Given  Key. 

In  addition  to  the  Dominant  chords  of  the  Related  keys  there  still 
remain  some  Common  chords  which  may  be  used  without  making  a 
transition. 

Taking  C  as  our  given  key,  the  related  key  of  F  possesses  two 
chords,  C7,  Z?b,  D  and  D,  F,  that  are  not  found  in  C.  The 
related  key  of  G  possesses  one  that  is  not  found  in  C, —  F,  D,  F$. 
See  accompanying  table  of  chords. 


K 

ey  of  C. 

</.         <r.  /. 

k 

K 

eyof  F.  a.  j 

( .     ^  i       X  |      /•  |  x 

b        _-J— ~d 

«_ 

K 

eyofG. 

»EI_f*  «  ^ 
i  1 

-—I     ,  d 

y  i 

m — id 

«g    f*-n£=-r    s=    *  -  1 

l-'  Tl 

The  use  of  these  chords  in  the  key  of  C  is  somewhat  restricted, 
owing  to  the  fact  that  they  are  not  owned  in  common  by  the  scale  in 
which  they  are  found  and  the  given  scale.  Their  use  generally 
indicates  a  transition  to  one  or  the  other  of  these  related  keys,  which 
is  continued  for  some  time.  The  following  examples  are  given  of 
their  use  without  a  transition  : 


44 


HARMONY. 


if* 


1HH 


None  of  them,  especially  3,  sound  quite  at  home. 

The  Major  and  Minor  scales  that  begin  on  the  same  keynote  are 
called  Parallel  scales.  The  chords  belonging  to  a  Minor  scale  may 
be  used  in  its  Parallel  major.  The  Minor  scale  (harmonic)  differs  from 
its  parallel  major  in  two  of  its  degrees  only;  viz.,  the  third  and  sixth, 
which  are  minor,  counting  up  from  the  keynote. 

n  Third.  Sixth 

These  sounds  are  found  in  three  of 
the  chords  in  the  minor  scale,  thus  : 


■i9- 


One  of  them,  the  sixth,  is  also  found  in  the  Diminished  chord  on 


:2s: 


1 


the   Supertoni  C    (which  must  be  used,  as  already  explained 
in  its  first  inversion)  ,  thus  : 

The  Minor  Tonic  and  Subdominant  are  often  preceded  by  the 
major  form  of  the  same  chord  (1)  ;  or,  the  Minor  Tonic  by  the 
Dominant  (2)  ;  the  Minor  Subdominant  by  the  Tonic  (3),  or  by 
the  Dominant  (4). 

1.  2.  3.  4. 


The  chord  on  the 
sixth  of  the  Parallel 
minor  may  be  pre- 
ceded by  the  Tonic  or 
Dominant. 


jGL 


is 


i 


HARMONY. 


45 


The  chord  on  the  sixth  of  the 
Major  may  be  followed  by  the 
Minor  Subdominant  or  Tonic. 


1— t 


I 


The  Minor  Tonic  may  also  be 
preceded  by  the  Dominant  of  the 
Relative  minor  (i)  ;  and  by  the 
Dominant  of  the  Relative  minor 
of  the  first  major  relation  (2). 


li 


The  first  inversion  of  the  dimin- 
ished chord  on  the  supertonic  is 
generally  followed  by  second  in- 
version of  tonic ;  it  may  be  pre- 
ceded by  any  chord  but  the 
mediant. 


m 


1 


The  foregoing  examples  show  what  may  follow  as  well  as  precede 
these  Parallel  Minor  chords.  The  pupil  should  try  the  effect  of  all  of 
these  successions  in  different  positions  and  inversions  (in  all  the  keys) . 
It  will  be  found  that  some  progressions  which  sound  well  with  one 
arrangement  of  the  chords,  are  intolerable  with  another  arrangement. 

The  last  Common  Chord  is  a  Major  chord  on  the  Lowered  Super- 
tonic  of  the  scale. 

In  the  key  of  C,  the  Lowered  Supertonic  is  Z?i? ;  a  Major  chord  on 
this  root  is,  Db,  jF,  Ab.  This  chord,  although  used  freely  in  the  Ma- 
jor key,  seems  to  sound  more  at  home  in  the  Minor  key.  If  used  in 
the  Major  key  it  may  be  preceded  by  the  Tonic,  Subdominant,  Domi- 
nant or  by  any  of  the  chords  of  the  Parallel  minor.  It  is  followed  by 
the  Tonic  or  Dominant. 


46 


HARMONY. 


This  chord  is  more  frequently  used  in  its  first  inversion  (as  in  ex- 
amples 4,  5,  6,  7)  and  is  then  called  the  Neapolitan  Sixth. 

In  the  example  that  follows,  all  the  chords  and  progressions  so  far 
learned  will  be  found.  The  analysis  of  it  should  be  well  studied. 
All  analyses  should  be  made  on  the  following  principles. 

I.  Every  chord  bears  some  relation  to  the  principal  key ;  that  is,  it 
is  either  to  be  found  in  that  key  or  in  one  of  its  relations. 

II.  Every  chord  bears  some  relation  to  the  Chord  that  precedes  it. 

III.  Every  chord  bears  some  relation  to  the  Chord  that  follows  it. 

IV.  Every  Major  chord  may  bear  five  possible  relations;  viz., 
Tonic,  Subdominant,  Dominant,  sixth  of  a  Minor  scale,  Lowered 
Supertonic. 

V.  Every  Minor  chord  may  bear  five  possible  relations;  viz., 
Tonic,  Subdominant,  Supertonic,  Mediant,  Submediant.  Thus, 
the  Major  chord  C,  E,  G,  in 

1,  bears  the  relation  of  Tonic  to  the  preceding  chord. 

2,  bears  the  relation  of  Dominant  to  the  following  chord. 

3,  bears  the  relation  of  Subdominant  to  the  following  chord. 

4,  bears  the  relation  of  Submediant  to  the  following  chord. 

5,  bears  the  relation  of  Lowered  Supertonic  (of  B,  followed  by 
dominant) . 


2. 

3- 

4- 

5- 

-i 

w  & — 1 

X 

The  Minor  chord  A,  C,  I?,  in 

I,  bears  the  relation  of  Tonic  to  the  preceding  chord. 


HARMONY. 


47 


2,  bears  the  relation  cf  Subdominant  to  the  following  chord. 

3,  bears  the  relation  of  Supertonic  to  the  following  chord. 

4,  bears  the  relation  of  Mediant  to  the  following  chord. 

5,  bears  the  relation  of  Submediant  to  the  preceding  chord. 

r.  2.  3.  4-  5- 


A  Chord  always  bears  a  relation  to  the  Chord  that  follows  it  differ- 
ing from  that  it  bears  to  the  Chord  that  precedes  it  (unless  the  same  chord 
precedes  and  follows) . 

This  relationship  may  or  may  not  make  a  modulation. 

1.  The  chord  A,  C,  E,  enters  as  Tonic,  and  is  followed  by  Sub- 
dominant  chord,  —  all  three  in  A  minor 

2.  A,  C,  E,  enters  as  Tonic,  but  is  Subdominant  to  the  following 

chord,  modulating  to  E  minor. 

3.  Dominant  of  C,  followed  by  chord  on  sixth  of  parallel  minor. 
This  chord,  A\>,  C,  E\>,  is  then  treated  as  the  Lowered  Supertonic 
of  G,  followed  by  Dominant  of  G. 

'■'•<' "  1.  2.  3.  mi^M 


— 

-h 

-id  

^= 

— <s — 

H  1  r 

 1  1  

- 

II 


Common  chords  have  been  treated  at  great  length  and  with  especial  fullness,  because  a 
thorough  familiarity  with  them  and  their  progressions  is  by  far  the  most  important  part  of  har- 
mony. The  dissonant  chords  that  follow  will  be  found  easy  if  the  common  chords  have  been 
thoroughly  mastered. 


I.  2. 


P 


3.     4.    5.  6. 


9.  10.   n.  12.    13.  14.  15. 


±=t== 

1*9 

4. 

r  - 

48 


HARMONY. 


1 6.     17.     l8.    19.    20.    2F.     22.    23.     24.  25.      26.    27.     28.    29.  30. 


9fc 


31.  32.    33.  34.35.    36.  37.    38.    39.   40.   41.   42.  43.    44.  45. 


r— t 


1  1 — L- f— J — p-  -p^ — ^ — % 


<s*-  -<^-  -IS- 


1.  Tonic.  2.  Tonic  parallel  minor.  3  and  4.  Dominant.  5. 
Submediant.  6.  Supertonic.  7  and  8.  Dominant  A  minor.  9  and 
10.  Tonic  parallel  minor.  11  and  12.  Submediant  parallel  minor. 
13.  Lowered  Supertonic  followed  by  (14)  dominant.  15.  Tonic 
16.  Submediant  is  Subdominant  to  the  next  chord.  17.  Tonic  E 
minor.  18.  Dominant  E  minor.  19.  Tonic  E  minor.  20.  Sub- 
dominant  E  minor,  but  mediant  to  the  next  chord  (21)  which 
is  dominant  to  (22)  Z?b,  which  is  lowered  supertonic  of  A,  followed 
by  dominant  (23)  and  (24)  tonic  A,  treated  as  subdominant  to  E, 
25.  26.  Dominant  of  E  minor,  followed  by  chord  with  root  a 
minor  third  above,  27.  28.  Dominant  of  D  minor,  followed  in  the 
same  way.     29.   C  minor,  parallel   minor.     The  two  bars  make 


HARMONY. 


49 


a  sequence.  30.  Dominant.  31  to  38.  All  belong  to  parallel 
minor,  except  35,  which  belongs  to  both  major  and  minor.  39  to  42. 
Are  easily  "parsed."  43.  Subdominant  of  parallel  minor,  preceded 
by  Submediant  of  major.  44.  Tonic  treated  as  sixth  of  E  minor. 
The  rest  is  easy. 

It  will  be  seen  that  the  chords  of  the  parallel  minor  may  be  written 
in  succession. 


Exercises  Introducing  All  Possible  Common  Chords  Without 
Going  Outside  of  the  Group  of  Related  Keys. 

I. 


1=3 


p=t 


P 


-in  1  n  r  - 


i 


11. 


in. 


St 


1= 


22: 


IV. 


sir 


-[  1  H  -2     — = 


5 


=2  - 


1 


V. 


Questions  on  Chapter  XL 

How  many  chords  are  common  to  the  scales  of  C  and  F  ?    Which  are  they  ? 

How  many  to  the  scales  of  C  and  G  ?    Which  are  they? 

Are  there  any  chords  common  to  all  three  scales?    Which  are  they? 

What  chords  in  F  are  not  found  in  C  ? 

What  chords  in  G  are  not  found  in  C? 

What  is  meant  by  parallel  scales  ? 


5° 


HARMONY. 


In  what  respect  does  a  minor  scale  differ  from  its  parallel  major? 
In  how  many  of  the  chords  of  the  minor  scale  are  these  notes  found  ? 
Does  the  dominant  of  a  major  scale  differ  in  any  way  from  that  of  its  parallel 
minor  ? 

How  is  the  minor  tonic  preceded?    The  minor  subdominant?    The  chord  on 
sixth  ? 

What  other  chords  may  precede  the  minor  tonic?  The  minor  subdominant? 
How  is  the  diminished  chord  on  supertonic  of  parallel  minor  generally  used? 
Is  any  other  common  chord  possible?    Where  is  its  root?    Is  it  major  or 

minor?    By  what  is  it  followed  ?    By  what  may  it  be  preceded  ?    In  what 

form  is  it  most  frequently  used  ? 
What  name  does  it  bear  in  this  case? 
How  many  relations  may  a  major  chord  bear? 
How  many,  a  minor  chord  ? 


HARMONY. 


5* 


CHAPTER  XII. 


Chord  of  Dominant  Seventh,  First  Progression. 


Dissonant  chords  are  those  that  include  a  sound  that  must  move  in 
some  specific  direction.    This  motion  is  called  the  Resolution  of  the 

Dissonance.  (The  term  resolution  is  rather  loosely  applied  to  the  movement  of  chords. 
Chords  progress,  their  Dissonant  members  resolved) 

All  Dissonant  chords  are  formed  by  additions  to  Major  Chords. 
The  most  important  are  the  additions  to  the  Dominant  chord. 

Dissonances  that  are  members  of  a  chord  are  called  Essential  (to 

distinguish  them  from  those  that  are  merely  ornamental,  or  that  are  used  as  Suspensions  or  Re- 
tardations). 

The  whole  series  of  dissonances  that  may  be  sounded  with  a  given 
root  are  called  its  Harmonies. 

The  first  addition  to  the  Dominant  is  the  seventh  over  the  root. 

This  seventh  is  minor;  it  resolves  by  descending. 

Three  Inversions  may  be  made.  The  rules  about  second  inversion 
of  common  chords  do  not  apply  to  Dissonant  chords. 

The  Dominant  seventh  chord  may  have  Three  Progressions. 

The  First  progression,  the  most  natural,  is  to  the  Tonic  Chord. 

Rides.    The  seventh  descends  one  degree. 

The  fifth  may  ascend  or  descend;  it  generally  descends. 

The  third  must  ascend  one  degree. 

The  root  when  at  the  bass  ascends  a  fourth  or  descends  a  fifth,  to 
the  root  of  the  tonic.    When  the  chord  is  inverted  the  root  is  stationary. 

It  will  be  seen  that  the  freedom  of  movement  of  common  chords  is 
lost  when  a  dissonant  is  added,  for  the  reason  that  they  must  move  to 
a  chord  of  which  the  note  that  the  dissonant  resolves  on  is  a  member. 

First  inversion. 

.  ...    .  L  or  ! 


5^ 


HARMONY. 


Second  inversion. 

— =1 


Third  inversion. 


Every  possible  arrangement  of  this  progression  is  given  here. 
Write  in  the  same  way  the  dominant  seventh  chords  of  the  keys  re- 
lated to  C,  taking  care  to  insert  the  necessary  accidentals  in  every 
case. 


Observe  I,  that  when  the  Root  is  at 
the  bass,  the  fifth  of  the  Tonic  has  to  be 
omitted.  This  is  often  avoided  by  omit- 
ting the  third  of  the  dominant  and  repeating 
the  Root,  thus : 


hi 


7^- 


3     I  I 


 £  1  LI 


or,  by  omitting  the  fifth,  etc.,  (2). 

II.  The  fifth  is  often  made  to  ascend,  when  in  the  soprano  or  at 
the  bass.  When  at  the  top  in  the  third  inversion,  it  may  ascend  to 
the  third  or  to  the  fifth  of  the  tonic. 

III.  When  the  Root  of  the  dominant  is  at  the  top  in  the  third 
inversion,  it  may  ascend  to  the  root  of  the  Tonic,     (it  might  do  so  when  at 

the  top  in  the  other  inversions,  but  does  not  sound  so  well.) 

The  introduction  of  this  additional  member  of  the  dominant  chord 
adds  very  much  to  our  resources  in  harmonizing  a  melody,  as  any 
note  in  the  melody  may  be  harmonized  as  the  seventh  of  a  dominant, 
provided  it  is  followed  by  the  note  on  the  next  degree  below,  and 
provided  it  will  be  the  seventh  in  one  of  the  dominant  chords  of  the 
Related  Group. 

2. 


1 


HARMONY.  53 

In  the  above  descending  scale,  harmonized  twice,  will  be  found  all 
the  dominant  seventh  chords  of  the  group  of  C,  and  its  relations,  ex- 
cept one;  viz.,  the  dominant  of  J?,  the  seventh  of  which  is  B\>. 
Observe  that  the  third  and  seventh  are  the  only  notes  in  the  scale 
which  may  not  be  sevenths. 

The  Bb,  needed  to  make  the  seventh  in  the  dominant  of  F,  makes 
the  fourth  chromatically  lowered  note — (the  others  being,  two  belonging  to 

the  parallel  minor,  and  the  lowered  supertonic)  . 

For  the  present  the  lowered  leading  note  of  the  scale  must  be  har- 
monized as  a  dominant  seventh. 


Analyze  the  following  example  before  writing  the  exercises  that 
follow. 


Proceed  thus  in  analyzing : 

First.    Chord  is  tonic  of  key,  quint  position. 

Second.  Dominant  of  A  minor,  second  inversion,  third  at  top, 
followed  by 

Third.  Tonic  of  A  minor,  octave  position,  mediant  to  the  chord 
following. 

Fourth.  Dominant  of  JR  major,  second  inversion,  seventh  at  top, 
followed  by 

Fifth.    Tonic  of  7%  tierce  position. 

Sixth.  Dominant  of  D  minor,  second  inversion,  seventh  at  top, 
followed  by 

Seventh.    Tonic  of  D  minor,  tierce  position. 
If  any  sequences  are  found  point  them  out. 


54 


HARMONY. 


ii. 


EE3 


III. 


«J  ' 


IV. 


Si? 


"HE* 


Ms- 


13 


 b 


z2 


PHIEI 


VI. 


—A-     k     4  -MM-    _    _                          r-     -1  1        r,  1  k  1               _          1^2-  -^=5      _            ,   -k.- 

— 1, — H — 

±3=1= 

VII. 

s 


There  are  a  few  exceptional  cases  in  which  the  seventh  "does  not 
descend  in  the  first  progression.  They  can  only  occur  with  the 
chords- arranged  exactly  as  follows. 

"  US 


IE!  ^  j2_ 

-©»  

r — r — r  ■ 

3 


-<S>-  -fS?- 


(Si  ~~  ^) 


7 


;ezee 


ii 


HARMONY. 


55 


r.  The  bass  and  top  part  ascending  in  thirds  with  each  other. 

2.  Root  at  bass  going  to  third,  and  seventh  ascending  to  fifth. 

(This  is  more  common  in  the  older  composers  than  it  is  now.) 

3.  Dominant,  tierce  position,  followed  by  first  inversion  of 
Snbdominant. 

4.  Diatonic  scale  in  contrary  motion. 

5.  Same,  but  the  seventh  is  doubled;  one  ascends,  the  other 
descends. 

Questions  on  Chapter  XII. 

What  is  a  dissonant  chord  ? 
What  is  this  movement  called? 
How  are  dissonant  chords  formed  ? 

Which  is  the  most  important  chord  to  which  they  may  be  added  ? 
What  is  an  essential  dissonance? 

What  name  is  given  to  the  series  of  dissonances  that  may  be  added  to  a  given 
root  ? 

What  is  the  first  addition  to  the  dominant  chord  ? 
Of  what  kind  is  this  seventh? 
How  does  it  resolve? 

How  many  inversions  may  be  made  of  a  dominant  seventh  chord? 
How  many  progressions  may  it  have? 
What  is  the  first  progression? 

How  do  the  members  of  the  dominant  seventh  move  in  this  progression  ? 
When  may  the  fifth  of  the  dominant  ascend  ? 

When  the  root  of  the  dominant  seventh  is  at  the  bass,  which  member  of  the 

tonic  following  it  must  be  omitted? 
How  may  this  omission  be  avoided? 
When  may  a  note  in  the  melody  be  a  seventh  ? 
How  many  and  which  notes  in  the  scale  may  be  sevenths? 
How  must  the  chromatically  lowered  leading  note  be  harmonized  ? 
How  many  notes  in  the  scale  may  be  lowered  accidentally? 
Which  are  they,  and  how  may  they  be  harmonized  ? 


56 


HARMONY. 


CHAPTER  XIII. 


Dominant  Seventh,  Second  and  Third  Progressions, 
and  Succession. 

A  dominant  seventh  chord  may  be  repeated  indefinitely,  with 
changes  in  its  position  and  inversion,  provided  that  the  dissonance 
is  resolved  when  the  Progression 


s*g- 


1 


4=t 


takes  place.  In  the  following 
example,  the  Dominant  chord 
is  struck  four  times  before  the 
progression  to  the  Tonic  takes 
place. 

The  Second  Progression  of  the  dominant  seventh  is  to  the  Sub- 
mediant  (sixth  of  scale) . 

This  progression  only  takes  place  with  the  Root  of  the  Dominant 
at  the  bass. 

The  Seventh  and  fifth  descend.  (The  fifth  must  descend,  otherwise  it  would 
make  parallel  fifths  with  the  root.) 

The  Third  ascends  except  when  the  fifth  of  the  dominant  is  at 
the  top.    When  this  is  the  case  it  may  descend. 
The  Root  ascends  to  the  root  of  the  submediant. 


i.  2.  3.  4.  5.  6. 


1,  3,  5.  Dominant  seventh,  followed  by  submediant. 

2,  4,  6.  Dominant  seventh,  followed  by  submediant  of  parallel 
minor. 


HARMONY. 


57 


In  5  the  third  descends,  the  fifth  of  the  dominant  being  at  the  top. 

Same  in  6.  This  sounds  better  in  instrumental  than  vocal  music, 
owing  to  the  awkward  skip  from 

Third  Progression.  The  dominant  seventh  of  a  Major  Key  may 
be  followed  by  the  Dominant  of  its  Relative  Minor. 

This  progression  may  take  place  with  any  inversion. 

The  seventh  descends;  the  fifth  and  third,  being  common  to  both 
chords,  either  remain  stationary,  or  they  may  change  places ;  that  is, 
while  cne  voice  moves  from  the  third  to  the  fifth,  another  may  reverse 
this,  moving  from  the  fifth  to  the  third. 

The  Root  ascends  chromatically. 

The  seventh  of  the  first  chord  may  skip  to  the  seventh  of  the 
second,  while  the  fifth  ascends  to  the  Root ;  or  the  seventh  may  skip 
to  the  fifth  of  the  second  chord ;  last,  the  seventh  may  be  omitted 
from  the  second  chord. 


6.  7.  8.  9.  10. 


1,  2,  3,  4.  Third  progression  with  root  at  bass  and  the  three  in- 
versions. 

5  and  6.  Third  and  fifth  changing  places. 

7.  First  inversion  of  one  followed  by  second  inversion  of  the  other. 

8.  Seventh  of  first  chord  skipping  to  seventh  of  second  chord. 

9.  Seventh  of  second  chord  omitted. 

10.  Seventh  skipping  to  fifth  of  second  chord. 

Note.  More  will  be  said  about  this  third  progression  in  the  chapter  on  chord  of  ninth. 


58 


HARMONY. 


It  is  necessary  to  observe  that  this  progression  may  only  be  used 
with  melodic  passages  that  will  permit  of  the  resolution  of  the 
second  chord  as  indicated  by  the  small  notes. 

In  addition  to  the  Three  Progressions  of  the  dominant  seventh 
chord,  it  is  possible  to  make  a  Succession  of  dominant  seventh  chords, 
as  follows : 

The  third  of  the  chord,  instead  of  being  made  to  ascend,  is  lowered 
chromatically,  and  is  made  the  seventh  in  the  succeeding  chord. 

This  Succession  has  the  root  and  fifth,  or  third  and  seventh  alter- 
nately at  the  bass ;  or  all  the  chords  may  have  the  roots  at  the  bass 
by  omitting  the  fifth  from  every  alternate  chord. 


-is>-k«>- 

2. 

■  ^  ^  ~&  & 

tr  h  | 

r-  -5>  

1  1 
X 

ISP 

a.       (22  |_ 

— ^  

22  - 

1.  Root  stationary,  fifth  descends,  root  and  fifth  at  bass. 

2.  Root  stationary,  fifth  descends,  third  and  seventh  at  bass. 

3.  Root  at  bass  goes  to  root  of  next  chord;  the  fifth  is  omitted 
from  chords  marked  x. 

This  succession  should  not  be  continued  for  more  than  two  or  three 
chords,  as  it  soon  grows  monotonous. 


Questions  on  Chapter  XIII. 

What  is  the  second  progression  of  dominant  seventh? 

How  do  the  members  of  the  chord  move? 

Which  member  of  the  dominant  must  be  at  the  bass? 

Does  the  third  always  ascend? 

What  is  the  third  progression  ? 

Can  the  dominant  of  a  minor  key  have  the  third  progression? 
How  do  the  members  of  the  chord  move  in  this  progression? 
What  is  it  necessary  to  observe  when  using  this  progression? 
How  is  a  succession  of  dominant  sevenths  made? 
How  do  the  other  members  of  the  chord  move? 

What  members  of  the  chord  may  succeed  each  other  as  bass  notes  in  this  suc- 
cession ? 

What  must  be  omitted  when  the  notes  are  at  the  bass? 


HARMONY. 


59 


Exercises  for  all  the  Progressions  and  the  Succession  of 
Dominant  Sevenths. 


I. 


mm 


III. 


#sb-4       f  i*  » 

r  1 

X 

• 

1^ 


!I1 


IV. 


X  X 


t4= 


Use  third  progression  at  the  signs  x  x. 


6o 


HARMONY. 


CHAPTER  XIV. 

Dominant  Ninth. 

The  next  addition  to  the  dominant  chord  is  the  Ninth  over  its  root. 

The  ninth  may  be  either  Major  or  Minor. 

Major  ninth  contains  an  octave  and  a  whole-tone. 

Minor  ninth,  an  octave  and  a  half-tone. 

In  Major  keys  both  kinds  of  ninths  are  used. 

In  Minor  keys  the  minor  ninth  only  may  be  used. 

The  ninth  resolves  by  descending  one  degree. 

The  ninth  may  not  be  used  as  a  bass. 

The  ninth  may  not  be  written  close  to,  or  below  the  root;  hence 
three  inversions  only  may  be  made  of  this  chord. 
'The  fifth  must  ascend  when  below  the  ninth. 

In  four-part  harmony,  the  third,  fifth,  or  seventh  may  be  omitted. 
The  first  progression  only  is  possible  when  the  ninth  is  added. 
The  Succession  of  dominant  chords  may  be  made,  provided  the 
ninth  is  used  only  with  alternate  chords. 

In  general,  the  minor  ninth  sounds  better  than  the  Major  when  it 


}t  at  the  top  of  the  chord, 
i.                2.              3.               4.            5.  6. 

7- 

U    1  f~ 
•    a   1  . 

r       r  r 

r  r 

T — r — &  TT 

<s>     a  \& — _L 

^  T  ^- 

:p  ^|=t-^^:= 

Third.  Fifth.  Seventh. 


8.  9.         10.        11.  12.  13.  14. 


HARMONY. 


61 


1.  Dominant  of  C,  with  major  ninth. 

2.  Dominant  of  C,  with  minor  ninth. 

3.  First  inversion.    4.  Second  inversion.    5.  Third  inversion. 

6,  7,  8.  The  third  omitted  in  6  and  7,  the  ninth  is  minor,  not  being 
at  the  top  of  the  chord. 
9,  10.  The  fifth  omitted. 
11,  12,  13.  The  seventh  omitted. 

14.  Succession  of  dominants.  The  ninth  is  not  added  to  the  sec- 
ond chord,  but  is  to  the  third.  If  the  succession  were  continued  the 
fourth  chord  would  be  without  the  ninth.  This  example  should  be 
written  in  several  keys. 

We  found  (  page  56  )  that  it  is  possible  to  change  the  arrangement 
of  a  chord  at  will,  and  that  there  was  no  Progression  until  the  har- 
mony changed;  but  as  the  ninth  in  descending  moves  to  the  root  of 
the  chord,  there  may  be  Resolution  without  Progression,  thus : 


The  ninth  and  seventh  may  also  move 
freely  to  any  other  member  of  the  chord  be- 
fore  progression  takes  place. 


1.  Ninth  leaps  to  seventh,  then  seventh  to  fifth. 

2.  Ninth  ascends  to  third. 

3.  Ninth  leaps  up  to  seventh. 

4.  Seventh  leaps  up  to  ninth. 

This  shows  that  resolution,  although  it  may  (  in  the  case  of  the 
ninth  )  take  place  without  progression,  is  not  compulsory. 


62 


HARMONY. 


The  chord  of  the  ninth  is  much  used  with  the  Root  omitted.  The 
remaining  notes  may  be  inverted  in  any  form,  and  may  have  the  first 
and  third  Progressions. 

The  omission  of  the  root  from  a  chord  with  Minor  ninth  makes 
it  what  is  known  as  the  Diminished  Seventh  Chord. 

The  fifth  may  descend  in  the  diminished  seventh  chord  when  it  is 
below  the  ninth,  if  the  ninth  is  not  at  the  top. 


i.  2.  3.  4.  5. 


1.  Third,  fifth,  seventh,  major  ninth,  first  progression. 

2.  Third,  fifth,  seventh,  major  ninth,  third  progression. 

3.  Third,  fifth,  seventh,  minor  ninth,  called  diminished  seventh 
chord,  because  the  interval  from  third  to  minor  ninth  is  a  diminished 
seventh. 

4.  Diminished  seventh  from  dominant  of  C,  and  diminished  seventh 
from  dominant  of  A,  its  relative  minor.  Observe  that  three  of  the 
letters  are  common  to  both  chords,  and  that  the  remaining  sounds  are 
enharmonically  identical.    It  is  owing  to  the  fact  of  these  two  chords 

(viz.,  the  dominant  of  any  key,  and  the  dominant  of  its  relative  minor)  having  SO  many 

sounds  in  common,  that  it  is  so  easy  to  pass  from  one  to  the 
other. 

5.  Z>,  the  fifth,  is  at  the  bass  and  descends  because  ^4t>,  the  ninth, 
is  not  at  the  top. 

Some  more  examples  of  the  third  progression. 

1.  First  chord  has  seventh  ;  the  second,  ninth  without  root. 

2.  Both  chords  have  the  ninth  without  root.  Of  course,  in  both 
these  cases  the  notes  that  are  common  to  both  chords  may  change 
places  in  a  variety  of  ways. 

3  and  4.  Give  examples  of  the  reversal  of  this  progression.  It  is 
possible  with  other  arrangements  of  the  chords,  but  is  most  effective 
in  the  two  here  given. 


HARMONY. 


i 


The  study  of  the  following  example  will  give  an  idea  of  the 
wealth  of  harmonic  possibilities  there  are  in  a  given  key  and  its  re- 
lations, and  we  are  by  no  means  at  the  end  of  them  yet. 

Take  as  given  key  C;  relative  majors -/FJ  G\   relative  minors  A, 


£>,  E. 


2. 


3.     -S-oife.  5-  -^-ks?- 


'9— 


IS 


12. 

7.  8.  9.  IO.     IT.  fo^L     3'  j?<2_       14.  15. 

„  piliiMiiii  ii^pi 

1.  The  natural  chords  of  the  scale. 

2.  Chords  from  parallel  minor. 

3.  Chord  on  lowered  supertonic  (during  a  modulation  into  a  re- 
lated key,  its  parallel  minor  and  lowered  supertonic  may  be  used.) 

4.  Dominant  harmony  of  the  key. 

5.  The  three  groups  derived  from  it. 
6  and  7.  Same  in  key  of  E. 

8  and  9.  Same  in  key  of  G. 
10  and  11.  Same  in  A  minor. 
12  and  13.  Same  in  D  minor. 
14  and  15.  Same  in  E  minor. 

Remember  especially,  that  in  using  these  dissonant  chords  no 
transition  is  made,  because  their  Tonics  are  the  Natural  Chords  of  the 
Given  Key.  Observe  also,  that  though  there  are  several  ways  of 
harmonizing  the  chromatically  lowered  notes,  they  may  all  be  ninths. 


64 


HARMONY. 


It  would  be  time  well  spent  to  write  out  a  number  of  tables  like  the 
preceding,  taking  care  to  get  in  the  accidentals  correctly. 

Observe,  that  the  seventh  in  the  dominant  of  a  major  key  is  the 
ninth  in  the  dominant  of  its  relative  minor;  therefore  all  the  notes  in 
the  scale  that  may  be  sevenths  (Chapter  XII.)  may  also  be  ninths  on 
the  same  conditions. 

Analyze  the  example  that  follows  before  writing  the  exercises. 
Give  the  name,  root,  and  progression  of  every  chord. 


6>— 


<5>- 


IB 


1= 


:£s:  Bfe-g;  g-g©-  3^±§f&?fHgzg:  -^-^  H 


3 


m 


-1=2- 


II 


The  succession  of  dominants  may  be  made  with  diminished  seventh 
chords.  Mistakes  are  often  made  in  writing  this  succession  ;  but  by 
applying  the  simple  rule  that  the  lowered  third  becomes  the  seventh 
in  the  next  chord,  and  remembering  the  roots  of  these  chords  are 
the  dominants  of  the  related  group,  no  mistake  need  occur.  Observe 
that  after  writing  six,  all  the  dominant  chords  in  the  group,  if  you 
wish  to  continue  the  passage,  return  to  the  first  chord  —  it  being  enhar- 
monically  the  same  as  A,  C,  E\>,  Gl?,  the  chord  that  would  follow 
jB\),  D\>,  G,  according  to  the  rule.  In  other  words,  when  the 
lowering  of  the  third  will  give  a  chord,  the  seventh  of  which  is  outside 
of  the  related  group,  substitute  the  diminished  seventh  chord  belong- 
ing to  the  key  of  the  mediant. 


HARMONY. 


65 


4- 


9i 


X 


1 


53 


i  II 


(This  is  treated  at  length  not  on  account  of  its  beauty,  but  because  so  many  who  do  use  it 
write  it  wrongly,  overloading  it  with  accidentals  that  should  never  appear  in  the  key.) 

Diminished  seventh  chords  may  move  up  as  well  as  down  (i)  ;  the 
seventh  when  at  the  bass  may  fail  to  the  tonic  (2). 


:2s: 


I'- 


ll 

II 


Note.  All  examples  and  exercises  should 
be  played  as  well  as  written.  This  is  the  only 
way  to  learn  to  remember  the  sound  by  the 
sight,  an  absolute  necessity  to  the  composer. 


Exercises  Introducing  the  Ninth. 

Use  the  ninth  without  the  root  more  frequently  than  with  it. 
I. 


II. 


II 


in. 


=2. 


IB 


IV. 


F— F 


-0—90- 


66 


HARMONY. 


Questions  on  Chapter  XIV. 

What  is  the  next  dissonant  after  the  seventh  that  may  be  added  to  the  domi- 
nant chord  ? 
Is  the  ninth  major  or  minor  in  major  keys? 
What  is  it  in  minor  keys? 
How  does  the  ninth  resolve? 
May  the  ninth  be  used  as  a  bass? 
May  the  ninth  be  close  to,  or  below  the  root? 
How  many  inversions  may  be  made  of  the  ninth  chord  ? 
How  must  the  fifth  move? 
May  the  fifth  ever  descend? 

What  members  may  be  omitted  in  four-part  harmony? 
What  progression  may  the  ninth  chord  have? 

How  may  the  succession  of  dominant  chords  be  made  when  the  ninth  is 
added  ? 

What  kind  of  ninth  sounds  best  when  it  is  not  at  the  top? 
In  what  form  is  this  chord  most  frequently  found  ? 
What  progressions  are  possible  without  the  root? 
What  is  the  origin  of  the  diminished  seventh  chord? 
When  may  the  fifth  descend  when  below  the  ninth? 

How  many  letters  are  common  to  the  dominant  of  the  major  and  the  dominant 

of  its  relative  minor?    Which  are  they? 
What  sound  have  they  in  common?    What  member  is  it  in  each  chord? 
In  what  one  way  may  all  chromatically  lowered  notes  be  harmonized? 


HARMONY. 


67 


CHAPTER  XV. 

Chord  of  Dominant  Eleventh. 


The  eleventh  from  the  root  is  the  next  addition  to  the  dominant  chord. 
The  third  of  the  chord  must  be  omitted.* 

The  movement  of  the  eleventh  is  either  down  to  the  third,  or  up 
to  the  fifth. 

The  ninth  moves  with  it  or  after  it,  when  the  eleventh  descends ; 
with  it,  when  the  eleventh  ascends. 

The  remaining  members  of  the  chord  are  stationary,  therefore  no 
Progression  takes  place. 

3- 


2. 


5  r    r  ^t=  t= 

1.  Eleventh  and  major  ninth 
descending. 

2.  Eleventh  and  minor  ninth 
descending. 

3.  Eleventh  and  major  ninth 
ascending. 


The  major  ninth  may  be  changed  to  minor  before  descending  to 
the  root. 


1 


9; 


-2- 


S3 


1.  Major  ninth  descending 
after  the  eleventh. 

2.  Major  ninth  changed  to 
minor  before  descending  after 
the  eleventh. 

3.  Eleventh  stationary;  of- 
ten used  in  terminations. 

As  is  the  case  with  the  sev- 
enth and  ninth  chords,  so  with 
the  eleventh ;  leaps  may  be 
made  freely  from  one  member 
of  the  chord  to  another. 


*  Some  weighty  authorities  say  the  third  may  be  sounded  with  the  eleventh.  It  is  almost  too 
harsh  for  sensitive  ears. 


68 


HARMONY. 


The  eleventh  chord  is  much  more  used  without,  than  with  the  root. 
The  notes  that  remain,  viz.,  the  fifth,  seventh,  ninth,  eleventh,  may 
have  the  Three  Progressions  of  the  Dominant.    First  Progression : 

The  eleventh  is  stationary. 
The  ninth  descends. 

The  seventh  may  go  to  any  member  of  the  tonic  chord. 
The  fifth  must  ascend  a  second  or  a  fourth. 
The  fifth,  seventh,  or  ninth  may  be  at  the  bass, 
i.         2.        3.         4.  5.  6. 


Fifth.  Seventh.  Ninth. 

1.  Fifth  at  bass  ascends  a  second;  seventh  at  top  descends. 

2.  Fifth  at  bass  ascends  a  second  ;  seventh  at  top  ascends. 

3.  Fifth  at  Dass  ascends  a  fourth;  seventh  at  top  descends.  If  it 
•ascended  here,  the  tonic  would  have  no  third. 

4.  Seventh  at  bass  ascends  to  fifth  of  tonic  ;  fifth  at  top  ascends  a 
second. 

5.  Seventh  at  bass  descends  to  third  of  tonic;  fifth  at  top  ascends 
a  second. 

6.  Seventh  at  bass  descends  to  tonic  ;  fifth  at  top  ascends  a  fourth. 

7.  Seventh  at  bass  falls  to  root  of  tonic  ;  fifth  at  top  ascends  a  second. 

8.  Ninth  at  bass. 

Observe  that  all  through  C  the  eleventh  does  not  move,  as  it  is  the 
root  of  the  tonic. 

In  all  these  examples,  A,  the  ninth,  may  be  ^4b,  because  the  ninth 
may  be  major  or  minor  in  major  keys. 


Second  Progression.  (To  Sub- 
mediant,  chord  on  sixth.)  The 
eleventh  and  ninth  are  stationary  ; 
the  seventh  descends  one  degree ; 
the  fifth  ascends  one  degree  ;  the 
fifth  or  seventh  mav  be  at  the  bass. 


9± 


Fifth. 


Seventh. 


HARMONY. 


69 


Third  Progression.  (To  dominant  of  relative  minor.)  Owing  to 
the  number  of  sounds  common  to  these  two  chords  (Chapter  XIV.), 
this  progression  takes  several  forms ;  any  of  the  notes  may  be  at  the 
bass. 


II 


1.  Fifth  at  bass,  ascends  to  root  of  dominant  relative  minor;  elev- 
enth, ninth,  and  seventh  descend. 

2.  Fifth  at  bass,  stationary;  eleventh,  ninth,  seventh  descend. 

3.  Fifth  at  bass;  seventh  and  fifth  stationary ;  eleventh  and  ninth 
descend. 

4.  Fifth  at  bass;  ninth,  seventh,  fifth  stationary;  eleventh  de- 
scends ;  result  is  that  both  are  eleventh  chords. 

*This  chord  may  have  the  same  root  as  the  preceding,  in  which  case  it 
would  be  third  to  ninth. 

Write  this  progression  out  with  every  possible  arrangement  of  the 
notes ;  also  the  first  and  second  progressions  with  different  arrange- 
ments of  the  upper  notes. 

The  eleventh  and  ninth  may  descend  while  the  seventh  and  fifth  re- 
main (as  if  the  root  were  present),  or  the  fifth  may  ascend  to  the 
root.  In  this  last  case,  if  the  fifth  is  at  the  bass  and  ascends  to  the 
root  of  the  dominant,  the  eleventh  and  ninth  may  ascend. 


N.  B.  The  descent  of  a  fifth  is  always  the  same  as  the  ascent  of 
a  fourth. 


7o 


HARMONY. 


The  following  example  gives  in  one  view  all  the  dominant  har 
monies  so  far,  and  tells  how  they  may  be  distinguished. 

a.     r.     2.     3%    4.  5. 


a.  Dominant  of  C,  with  added 
notes  as  far  as  the  eleventh. 

Broken  into  five  groups  of  four 
sounds  each. 


HI 


First  group,  counting  from  lowest  note,  consists  of  major  third, 
perfect  fifth,  minor  seventh ;  is  therefore  Root  to  Seventh. 

Second  group  (counting  as  before)  consists  of  minor  third,  dimin- 
ished fifth,  minor  seventh ;  is  therefore  Third  to  Major  Ninth. 

Third  group  consists  of  minor  third,  diminished  fifth,  diminished 
seventh ;  is  therefore  Third  to  Minor  Ninth. 

Fourth  group  :  minor  third,  perfect  fifth,  minor  seventh ;  is  there- 
fore Fifth  to  Eleventh,  with  major  ninth. 

•  Fifth  group  is  like  2  ;  therefore  a  group  consisting  of  minor  third, 
diminished  fifth,  and  minor  seventh,  may  be  either  Third  to  Major 
Ninth,  or  Fifth  to  Eleventh,  with  minor  ninth.  To  decide  which  it 
is,  it  is  only  necessary  to  remember  what  the  roots  of  the  dominant 
chords  of  the  Related  group  are.  Therefore  Z>,  F,  A\>,  C,  if  found 
in  the  key  of  C,  must  be  fifth  to  eleventh,  with  minor  ninth,  because 
if  it  were  third  to  major  ninth,  it  is  evident  that  Ffo  would  be  the 
root. 

As  to  the  other  group,  B,  D,  F,  A,  it  may  be  third  to  ninth, 
dominant  of  C,  or  fifth  to  eleventh  in  the  dominant  of  the  relative 
minor  of  C. 

Arrange  the  following  groups  in  thirds,  and  find  the  roots.  State 
to  what  keys  they  belong,  and  write  their  progressions. 


?— a. 


I 


Questions  on  Chapter  XV. 

What  member  of  the  chord  must  be  omitted  when  the  eleventh  is  added  ? 
How  does  the  eleventh  move? 


HARMONY. 


What  member  moves  with  it? 

Does  the  ninth  always  move  with  the  eleventh? 

In  what  way  is  this  chord  generally  used  ? 

What  progressions  may  it  have  when  the  root  is  omitted  ? 

What  are  the  movements  of  the  notes  in  the  first  progression? 

What  notes  may  be  at  the  bass? 

Give  the  movements  of  the  notes  in  the  second  progression. 

What  may  be  at  the  bass  ? 

Give  movements  of  notes  in  third  progression. 

Which  notes  may  be  at  the  bass  ? 

May  eleventh  and  ninth  move  as  if  root  were  present? 

How  many  groups  of  four  sounds  each  may  be  derived  from  the  dominant 
harmonies? 

Of  what  intervals  does  the  first  group  consist?  The  second  ?  The  third  ?  The 

fourth?    The  fifth? 
Which  two  groups  are  alike  ? 

How  may  the  roots  of  these  similar  groups  be  determined  in  a  given  key  ? 
How  must  a  group  be  arranged  in  order  that  the  root  may  be  found  ? 


7' 


HARMONY. 


CHAPTER  XVI. 

Chord  of  Dominant  Eleventh,  Continued.  Additional 
Remarks  on  Second  Inversions. 


In  the  following  examples  will  be  found  the  passages  in  which 
the  eleventh  is  most  frequently  found. 

i.  2.  3.  4. 


1 .  Melody  moves  from  first  to  third  of  major  scale. 

2.  Melody  moves  from  third  to  fifth  of  major  scale. 

3.  Melody  moves  from  first  to  third  of  minor  scale. 

4.  Melody  moves  from  third  to  fifth  of  minor  scale. 

5.  Melody  ascends  from  fifth  of  dominant  to  fifth  of  tonic. 

6.  This  passage  is  frequent  in  terminations.  In  the  last  bar  but 
one  the  seventh  ascends  instead  of  remaining  stationary. 

We  have  found  that  the  succession  of  dominants  might  take  place 
when  ninth  was  added  (  page  64  ) .  It  may  also  take  place  when 
eleventh  is  added. 


HARMONY. 


73 


6. 


i 


 1- 


s-^— fit 


"25^ 


=t=i= 


±= 


i 


1.  First  chord, seventh  ;  second,  yz/*^  eleve7tth. 

2.  First  chord,  third  to  ninth;  second,  fifth  to  eleventh. 

3.  First  chord,  third  to  minor  ninth  (  at  bass  )  ;  second,  fifth  to 
eleventh. 

4.  Both  chords,  fifth  to  eleventh. 

5.  First  chord,  fifth  to  eleventh  ;  second,  first  to  seventh. 

6.  First  chord,  fifth  to  eleventh  ;  second,  third  to  ninth. 
Observe  that  in  every  case  the  ninth  may  be  major  or  minor. 
All  these  progressions  may  be  reversed. 

The  rule  as  first  given  for  the  Succession  (  page  58  )  may  now  be 
expressed  as  follows,  since  the  succession  may  take  place  when  the 
third  is  absent. 

The  harmonies  of  two  or  more  roots  descending  by  fifths  may  be 
written  in  succession. 

_g   .       Harmonies  from   all  the 

t>s>g — kg— 1 1    dominants    of    the  related 
Roots.  B.      E.       A.    D.     G.      C.         group  in  succession. 


-M-- 


Additional  Remarks  on  Second  Inversions. 


The  second  inversion  of  a  chord  may  be  followed  by  any  chord 
with  the  same  bass  as  that  of  the  second  inversion,  or  with  a  bass  one 
degree  above  or  below  that  of  the  second  inversion.  The  second  in- 
version of  the  tonic  may  be  followed  by  the  first  inversion  of  the  tonic. 


i 


Eg: 


(2- 


74 


HARMONY. 


5- 


1.  Second  inversion  of  tonic,  followed  with  various  chords. 

2.  Second  inversion  of  subdominant,  followed  by  dominant. 

3.  Second  inversion  of  dominant. 

4.  The  chord  of  ^4  minor,  second  inversion,  enters  as  a  tonic; 
but  as  it  bears  the  relation  of  mediant  to  the  key  of  F,  advantage  is 
taken  of  the  fact  that  a  second  inversion  may  be  followed  by  any 
chord  with  the  same  bass,  and  the  dominant  of  F  follows  it. 

5.  Second  inversion  of  tonic,  changed  to  a  dominant  by  the  addi- 
tion of  the  seventh  to  which  the  fifth  skips. 


Exercises  for  the  Eleventh  Chord. 

In  the  first  two  the  places  are  marked  where  an  eleventh  chord 
may  be  used. 

'^J 


mm 


11. 


HARMONY. 


75 


III. 


IV. 


II 


V. 


at* 


i 


VI. 


FEE 


I 


76 


HARMONY. 


CHAPTER  XVII. 

The  Progressions  of  the  dominant  so  far  given,  are  the  most  natu- 
ral and  most  usual ;  but  successions  may  be  made  of  which  no  satis- 
factory explanation  can  be  given,  in  accordance  with  the  following 
rule : 

Any  two  of  the  Dominant  Harmonies  of  the  Related  Group  may 
be  written  in  succession  (with  a  few  exceptions) ,  provided  they  have  at 
least  one  sound  in  common. 

The  dissonant  notes  are  not  always  resolved  in  these  successions, 
nor  are  these  successions  agreeable,  except  in  certain  arrangements 
of  the  chords. 

The  first  thing  to  be  done,  is  to  find  in  which  dominant  harmonies 
each  note  in  the  scale  may  be  found  as  root,  third,  fifth,  seventh, 
ninth,  or  eleventh. 


Connecting  Note  C. 


Root. 


Seventh. 


Ninth. 


Eleventh. 


C  is  root  in  dominant  of  F\  seventh  in  dominant  of  G ;  ninth  in 
dominant  of  E  minor;  eleventh  in  dominant  of  C. 


Connecting  Note  D. 

Root.  Fifth.  Seventh.  Ninth.  Eleventh. 


D  is  root  in  dominant  of  G ;  fifth  in  dominant  of  C;  seventh  in 
dominant  of  A  minor ;  ninth  in  dominant  of  F\  eleventh  in  dominant 
of  D  minor. 


HARMONY.  7 
Connecting  Note  E. 

Root.  Third.  Fifth.  Ninth.  Eleventh. 


E  is  root  in  dominant  of  A  minor ;  third  in  dominant  of  F\  fifth 
in  dominant  of  D  minor ;  ninth  in  dominant  of  G ;  eleventh  in  domi- 
nant of  E  minor. 

Connecting  Note  F. 

Seventh.  Ninth.  Eleventh. 


-  1  #~1  J 

U  *  i  i  i  II 

F  is  seventh  in  dominant  of  C\  ninth  in  dominant  of  A  minor; 
eleventh  in  dominant  of  F. 


Connecting  Note  G. 


Root.  Fifth.  Seventh.  Eleventh. 


G  is  root  in  dominant  of  C;  fifth  in  dominant  of  F\  seventh  in 
dominant  of  D  minor;  eleventh  in  dominant  of  G. 


Connecting  Note  A. 

Fifth.  Seventh. 


mm 


Ninth.  Eleventh. 
X 


A  is  root  in  dominant  of  D  minor;  fifth  in  dominant  of  G ;  seventh 
in  dominant  of  E  minor;  ninth  in  dominant  of  C;  eleventh  in  dom- 
inant of  A  minor. 


Root. 

m 


Connecting  Note  B. 

Third.  Fifth.  F  \s  root  in  dominant  of  E 

minor ;  third  in  dominant  of 
C;  fifth  in  dominant  of  A 
minor. 


m 


78 


HARMONY. 


The  chords  marked  x  cannot  be  written  in  succession,  although  they 
have  a  connecting  sound. 

We  give  some  examples  below  of  these  successions,  in  the  arrange- 
ments which  are  most  effective.  The  pupil  should  try  other  arrange- 
ments and  should  write  them  in  various  keys.  No  other  exercise  can 
equal  this  in  giving  the  student  a  sure  grasp  of  the  possibilities  of 
chord  successions  without  transition. 

The  successions  that  come  under  rules  already  given  are  omitted ; 
viz.,  the  third  progression  (page  57),  and  the  succession  by  lower- 
ing the  third  (page  58.) 


Connecting  Note  C. 


N.B. 


N.B. 


1.  From  C,  E,  G,  .Z>b,  to  chords  derived  from  D.  This  will 
serve  as  a  model  for  all  successions  in  which  the  connecting  note  is 
the  root  of  the  first  group,  and  seventh  in  the  second  group. 

2.  Connecting  note  root  and  eleventh. 

3.  Connecting  note,  root  and  ninth. 

4.  Connecting  note  eleventh  and  seventh.  This  sounds  well  in  any 
arrangement. 

Connecting  Note  D. 

As  root  and  fifth,  any  arrangement,  succession  of  dominants.  As 
root  and  seventh,  see  Model  (i.  e.,  C,  E,  G,  B\>,  to  derivatives 
of  D). 


9- 


— 11 


1.  Root  and  ninth;  the  second  group  may 
be  either  third  to  ninth,  ox  fifth  to  eleventh. 
If  considered  as  third  to  ninth,  the  root  is 
dominant  of  E;  if  as  fifth  to  eleventh,  root 
is  dominant  of  the  relative  minor  of  E. 


HARMONY. 


79 


Connecting  Note  E. 


3Z- 


:5f-%l_: 


JS2  


9i— -F 


:t=: 


1 .  As  root  and  third. 

2.  As  root  and  ninth  or  eleventh  (root  of  second  chord,  D  or  B). 
As  root  and  fifth,  succession  of  dominants. 

Connecting  Note  F. 
As  seventh  and  ninth,  third  progression.    As  seventh  and  eleventh. 

Connecting  Note  G. 

As  root  and  fifth,  succession  of  dominants.  As  root  and  seventh. 
(  See  Model.) 

„     i.     N.B.  2. 


i — 


8  rg— 


ff — p§ — | 


I 


9^ 


-|2  12- 


1 


1.  Gives  an. example  of  the  reversal  of  the  rule  for  the  succession 
of  dominants;  i.  e.,  raised  seventh  becoming  third  when  the  root  of 
the  first  chord  is  present.  First  measure  gives  the  arrangement  that 
sounds  best. 

2.  As  root  and  eleventh. 


II 


II 


Connecting  Note  A. 

As  root  and  fifth,  succession  of  dominants. 
As  root  and  seventh.     (  See  Model.) 


Root.       Ninth  or 
eleventh. 


80  HARMONY. 

Connecting  Note  B. 

As  root  and  fifth,  succession  of  dominants, 
i. 


i 


iP=^=jLrp  ht^=y 


i 


i.  As  root  and  third. 

These  examples,  with  two  exceptions,  only  give  the  movements 
from  the  first  -chord  to  the  others  containing  the  same  connecting 
note  ;  but  the  movement  may  take  place  from  any  chord  of  the  group 
to  any  other,  with  scarcely  an  exception.  All  the  successions  given 
may  be  reversed. 

The  two  chords  marked  N.  B.  in  the  first  example  are  identical  in 
sound,  because  the  diminished  seventh  derived  from  the  dominant 
of  a  major  key,  and  that  derived  from  the  dominant  of  its  relative 
minor,  are  enharmonically  the  same,  thus : 

B,  D,  F,  ufi^G%  B,  D,  F 
Dominant  of  C,  dominant  of  A  minor. 

B,  G,  B\>,  J3vTc%  B,  G,  B\>. 
Dominant  of  F,  dominant  of  D  minor. 

F%  A,  C,  B^B^  Fl  A,  C, 
Dominant  of  G,  dominant  of  B  minor. 

The  minor  ninth  in  the  dominant  of  the  major  is  enharmonically  the 
third  in  the  dominant  of  its  relative  minor. 

Observe  in  all  these  successions  that  the  connecting'  note  must  re- 
main in  the  same  part. 

The  following  successions  in  which  there  is  no  connecting  note 
may  be  made. 


HARMONY. 


Si 


i  and  2.  The  second  chord  resolved  as  a  dominant;  in  2  the 
seventh  falls  to  the  root  of  the  tonic. 

3  and  4.  Diminished  seventh  chord  resolves  as  a  supertonic  har- 
mony (Chapter  XVIII.).  It  may  resolve  as  a  dominant, but  the  first 
is  more  agreeable. 

Exercises  for  Eleventh  Chord,  and  the  Unusual  Progressions 
Treated  in  the  Previous  Chapters. 

Eleventh  may  be  used  at  x.  Look  for  opportunites  for  some  of  the 
successions  given  in  last  chapter.  Where  two  notes  in  succession 
are  marked  x  x  there  is  an  opportunity. 

I. 


82 


HARMONY. 


IV. 


in 


XXX 


^  7rS~ 

=t==t 

X 

F — 

-U  i'. 

3 

Seventh.  Root. 


VI. 


See 


m 


HARMONY. 


83 


CHAPTER  XVIII. 
Chord  of  Thirteenth. 

In  the  majority  of  cases  the  thirteenth  may,  for  practical  purposes, 
be  treated  as  a  changing  note,  or  as  a  retardation.  Still  there  are 
some  cases  in  which  its  movement  is  such  that  it  must  be  looked  upon 
as  an  essential  dissonance. 

The  thirteenth,  like  the  ninth,  may  be  major  or  minor.  The  fifth 
is  generally  omitted  when  the  thirteenth  is  added,  especially  in  those 
cases  in  which  the  thirteenth  descends  to  the  twelfth  (i.  e.,  the  fifth 
of  the  chord). 


Thirteenth. 


<2- 


H 


Dominant  of  C,  with  all  the  available 
overtones  to  the  thirteenth. 


■>■  11 


In  the  examples  that  follow  the  cases  are  given  in  which  the 
thirteenth  may  be  looked  upon  simply  as  a  changing  note. 

1.  2.  3.  4.  5. 


r 

=P==^  E^=^  =F=^=  ±=^41 


1.  Root,  third,  seventh;       is  the  thirteenth,  or  a  changing  note. 

2.  Root,  seventh,  and  ninth.  3.  Third,  seventh,  and  ninth. 
4.  Seventh,  ninth,  and  eleventh.  A  may  be  flat  or  natural,  at  will. 
If  all  the  E's  are  made  flat,  the  examples  would  be  in  C  minor.  In 
this  case  Ab  would  be  necessary  also. 

5.  D\  is  a  convenient  mis-writing  for  E\>,  the  minor  thirteenth. 

(  See  remarks  on  augmented  fifths.) 


84 


HARMONY. 


[n  the  following  examples  it  is  not  possible  to  look  upon  it  as  a 
changing  note  or  retardation. 

6. 


5- 


IS. 


—i- 


-g>— 


rT 


5>t 


11 


7.  8.  9.  10.  11.  12. 

— ^r   1-  -4  -r—p  *   e  •  I  <i  ^   I  *      »    f  •  II 

i.  The  thirteenth,  falls  to  the  root  of  the  tonic.  2.  Same, 
second  progression. 

3.  Third  progression,  thirteenth  falls  to  seventh  of  dominant  of 
relative  minor. 

4.  This  looks  like  a  second  inversion  of  E  minor.  It  is  the  only 
case  in  which  a  group  of  notes  that  form  a  common  chord  must  be 
considered  as  a  dominant  group.  From  the  manner  of  its  entry  it 
cannot  be  a  second  inversion  of  subdominant  or  dominant;  and  the 
strong  upward  tendency  of  the  B  proves  that  it  is  not  a  tonic,  but  a 
piece  of  dominant  harmony. 

5.  Thirteenth  stationary,  seventh  and  ninth  ascending  to  eighth 
and  third. 

6.  Major  ninth,  eleventh,  major  thirteenth,  resolving  within  the 
chord. 

7.  Major  ninth,  eleventh,  minor  thirteenth,  resolving  on  tonic. 
Minor  thirteenth  ascends  chromatically.     (  In  this  case  the  £0  is  generally 

written  D$,  making  an  augmented  sixth  chord.) 

8.  Same  group  with  minor  ninth  and  thirteenth  resolved  on  tonic. 


HARMONY. 


85 


In  this  case  the  E\>  must  be  used,  as  the  combination  of  A\>  and  D\ 
would  be  a  monstrosity. 

9.  A  succession  of  dominants,  the  first  with  minor  thirteenth. 

10.  Same  group  as  8,  resolving  within  the  chord. 

11.  Dominant  with  minor  thirteenth,  followed  by  augmented  sixth. 

12.  First  chord,  dominant,  root,  third,  seventh,  ninth;  second 
chord,  dominant,  root  and  ninth  having  changed  places;  viz.,  bass, 
ninth;  tenor,  eleventh;  alto,  thirteenth;  soprano,  root.  This  passage 
is  the  only  one  in  which  the  root  may  be  struck  over  the  ninth  with 
good  effect,  owing  possibly  to  the  fact  that  the  whole  bar  consists  of 
dominant  harmony,  and  also  to  the  diatonic  contrary  motion. 

In  analyzing  harmony  a  good  rule  to  follow  is,  take  the  most  obvious 
explanation,  especially  when  the  combinations  may  be  ''parsed"  as 
common  chords,  instead  of  hunting  for  hidden  "roots."  For  exam- 
ple, the  following  passage  from  Tannhauser  is  thus  explained  by  one 
author.  The  note  marked  x  is  enharmonically  B\>,  consequently  is 
the  ninth,  C$  the  third,  and  F\  the  thirteenth  in  the  dominant  of  D. 
But  if  it  is  looked  on  simply  as  the  second  inversion  of  JF$  major, 
changed  to  F\  minor,  and  followed  by  the  chord  on  the  sixth  (  D 
major),  a  much  simpler  explanation  maybe  given;  viz.,  B  minor, 
the  preceding  chord,  is  the  minor  subdominant  of  F$,  which  enters 
as  a  tonic;  second  inversion  is  changed  to  minor,  and  followed  by 
the  chord  on  the  sixth. 

The  beauty  of  the  passage  is  owing  largely  to  the  uncertainty  of 
key  and  the  kaleidoscopic  rapidity  with  which  the  changes  of  key 
take'  place. 


The  following  passage,  also  by  Wagner,  the  same  author,  calls 
an  unusual  form  of  augmented  sixth,  calling  the  Lfo  a  miswriting 
for  Cj},  in  spite  of  the  fact  that  the  C  is  natural  when  the  harmony 
changes. 


S6 


HARMONY. 


It  is  much  easier  to  say  it  is  the  diminished  seventh  chord,  dom- 
inant harmony  of  Bfo.  The  Lfo  is  a  changing  note  which  leaps  to 
another  member  of  the  chord,  then  runs  to  its  resolution,  C.  The 
chord  A\,  C,  E\>,  Gb,  is  enharmonically  A,  C,  E^,  a  derivative 
of  the  chord  that  follows.  It  follows  that  in  both  instances  Wagner 
wrote  his  chords  properly. 


x 


WjP  *  *  

i— -— /- 

*  - 

J  ii 

-•- 

*  f — *— 

>  i 

Chord  of  eleventh,  dominant  of  Bb.         Eleventh  resolved. 


The  best  way  to  become  familiar  with  this  chord  is  to  transpose 
the  examples  into  all  the  keys. 


HARMONY. 


*7 


CHAPTER  XIX. 


Supertonic  Harmony. 


There  is  another  progression  of  dissonant  groups,  that  differs  so 
widely  from  any  that  they  have  as  Dominant  harmonies,  that  to  dis- 
tinguish them  they  are  called  Supertonic  Harmonies,  from  the  fact 
that  their  progression  is  to  the  Tonic  chords  of  the  keys  in  which  their 
Roots  are  the  Supertonics. 

The  major  ninth  is  stationary.  The  minor  ninth  ascends  chromat- 
ically in  Major  keys,  is  stationary  in  minor  keys.     (  The  major  ninth  cannot 

be  used  in  minor  keys. ) 

The  seventh  is  stationary. 

The  fifth  and  third  go  to  the  fifth  of  the  Tonic. 
The  root  when  at  the  bass  goes  to  the  fifth  of  the  Tonic  ;  when  not  at 
the  Bass,  to  the  third  of  the  Tonic  generally,  but  it  may  go  to  the  fifth. 

1.  Supertonic  harmony 
of  C,  with  major  ninth 
stationary. 

2.  Supertonic  harmony 
of  C,  with  minor  ninth 
ascending  chromatically. 

3.  Supertonic  harmony  of  A  minor,  with  minor  ninth  stationary. 
As  the  root,  third,  and  fifth  all  go  to  the  fifth  of  the  tonic,  it  will  at 
once  be  seen  that  the  second  inversion  of  the  tonic  generally  follows. 

1.        2.         3,  4.  5.  6.  7.  .  8. 


9 


1 .  First  inversion,  with  root  present. 

2.  First  inverson,  root  omitted,  major  ninth. 


88 


HARMONY., 


3.  First  inversion,  root  omitted,  minor  ninth. 

4.  Second  inversion,  root  present. 

5.  Second  inversion,  root  omitted,  major  ninth. 

6.  Second  inversion,  root  omitted,  minor  ninth. 

7.  When  the  root  is  at  bass  it  is  generally  repeated,  and  the  fifth 
is  omitted. 

8.  When  the  third  is  at  bass  and  the  ninth  is  minor,  the  third  may 
fall  to  the  root  of  the  tonic. 


1=1 


1.  When  the  seventh  is  at  the  bass,  it  is  generally  repeated  when 
the  ninth  is  at  top. 

2.  Minor  ninth,  fifth  at  top. 

3  and  4  give  the  usual  way  of  writing  these  two  chords;  viz., 

th  D%  instead  of  E\>. 

5.  Has  the  minor  ninth  at  bass. 

The  supertonic  harmony  may  be  preceded  by  the  eleventh 
chord. 


J-J- 


1 


ill 


Also  by  the  first  inversion  of  the  chord  on  lowered  supertonic. 
This  is  the  only  case  in  music  in  which  it  is  possible  to  write  in  suc- 
cession chords  arising  from  two  forms  of  the  same  root;  viz.,  Z?b, 
F,  A\>,  and  D%  F%,  A\. 


HARMONY,  89 


0      1  1 

1 

—&— 

— 1  

— 

_[ 

11= 

1 

:^|— 

-tr-Si  & 

-I 
— 1 

— 

— 

+  - 

= 

The  eleventh  of  the  supertonic  harmony  may  be  used.  It  sounds 
best  with  the  minor  ninth. 


1  and  2.  Fifth  at  the  bass. 

3.  Minor  ninth  at  the  bass. 

4.  Seventh  at  the  bass. 


Questions  on  Chapter  XIX. 

What  distinguishes  the  supertonic  harmony  from  the  dominant? 

How  does  the  major  ninth  proceed?    The  minor  ninth  in  major  keys?  The 

minor  ninth  in  minor  keys  ?    The  fifth?   The  third?    The  root,  when  at 

bass?    The  root,  when  not  at  bass? 
In  what  form  does  the  tonic  generally  follow  it? 
Which  members  of  this  chord  are  used  when  the  root  is  at  the  bass? 
Does  the  third  always  ascend  ?    When  may  it  not? 
Give  the  rule  for  use  of  seventh  as  a  bass. 
What  miswriting  is  often  used  in  this  case? 
May  the  eleventh  of  supertonic  be  used? 


Exercises  on  Supertonic  Harmony. 


When  the  second  inversion  of  tonic  follows  the  supertonic  har- 
mony, be  careful  to  follow  it  according  to  rule. 
I.  x 


az2= 


EEE 


90 


HARMONY. 


II. 


X  X 

nn 


in. 


22: 


tat 


5^ 


HARMONY.  91 


CHAPTER  XX. 

Altered  Chords,  Augmented  Sixth,  Augmented  Fifth, 
Passing  Seventh. 

There  are  a  few  chords  that  are  produced  by  altering  one  of  the 
members  of  a  given  chord,  chromatically. 

Strictly  considered,  this  altered  note  is  nothing  but  a  passing  note 
between  the  member  of  the  chord  and  the  note  to  which  this  mem- 
ber moves. 

The  most  important  of  these  chords  by  alteration  is  called  the  Aug- 
mented Sixth  Chord. 

This  Chord  results  from  the  chromatic  lowering  of  the  fifth  of  a 
dominant  or  supertonic  harmony,  root  to  seventh  or  third  to  minor 
ninth. 

The  progression  of  the  chord  is  unchanged.  It  maybe  inverted  in 
any  Way,  but  the  chromatically  lowered  fifth  is  generally  used  as  a 
bass. 

The  chord  gets  its  name  from  the  fact  that  the  interval  between 
the  lowered  fifth  and  the  third  of  the  chord  is  an  Augmented  Sixth  ; 
thus,  D,    Ah,  F% 

Root,   Fifth,  Third. 


1.  Progressing  like  a  dominant,  root  present. 

2.  Progressing  like  a  supertonic  harmony,  root  present. 

3  and  4.  Same  as  1  and  2,  but  the  root  is  omitted  and  minor 
ninth  added. 

5  and  6.  Progressing  like  a  dominant  to  a  minor  tonic.     This  pro- 


92 


HARMONY, 


gression  of  the  augmented  sixth  is  unusual.  When  it  progresses  like 
a  dominant,  the  chord  to  which  it  moves  is  (generally)  also  a 
dominant. 

The  parallel  fifths  that  occur  in  3  and  6  are  not  at  all  unpleasant. 
They  may  be  found  in  the  best  writers. 

The  following  inversions  of  the  augmented  sixth  are  used. 


— tr-=-t- 


9&- 


m 


1 .  The  upper  notes  may  be  arranged  in  any  way ;  the  seventh  of 
the  original  chord  is  at  the  bass. 

2.  Minor  ninth  at  the  bass. 

3.  Third  at  bass.  2  and  3  sound  best  when  arranged  as  here 
given. 

:  ,      r  ~;  .  || 


,  The  rule  for  the  succession 
of  dominant  chords  by  lower- 


ing the  third  chromatically 
applies  to  this  chord. 


II 


1.  Root  present.  2.  With  minor  ninth.  The  Eft  is  suspended  to 
avoid  the  parallel  fifths  between  the  outer  parts. 

The  minor  ninth  may  be  resolved  and  the  lowered  fifth  restored 
at  the  same  time. 


=t=: 


3- 


=t=A 


4= 


1.  The  ninth  at  top,  lowered  fifth  at  bass. 


HARMONY. 


93 


2.  The  ninth  at  bass,  lowered  fifth  at  top.  This  progression  is  of 
singular  beauty.  It  is  often  incorrectly  written  ;  viz.,  the  A\>  written 
GjJ,  or  the  F$  written  GP. 

3.  The  same,  but  with  the  third  lowered,  making  a  succession  to 
fifth,  seventh,  ninth,  eleventh,  dominant  of  C.  Then  the  ninth 
{A\>)  of  this  chord  is  resolved,  and  the  fifth  at  the  same  time 
lowered. 

Rewrite  the  exercises  on  supertonic  harmony,  substituting  an 
augmented  sixth  whenever  the  supertonic  harmony  was  used  (ex- 
cept with  the  notes  marked  x). 

Questions  on  Chapter  XX. 

How  is  the  chord  of  augmented  sixth  formed? 
What  progressions  has  it? 

Which  member  of  the  chord  is  generally  used  as  a  bass? 

Between  which  two  members  of  the  chord  does  the  interval  of  the  augmented 

sixth  lie  that  gives  the  chord  its  name? 
Is  the  succession  by  lowering  the  third  possible  with  this  chord? 

Chord  of  Augmented  Fifth. 

This  chord  may  be  produced  in  two  ways :  — 

First,  by  chromatically  raising  the  fifth  of  a  Major  chord. 

Second,  by  chromatically  lowering  the  root  of  a  Minor  chord. 

The  raised  fifth  must  ascend ;  any  chord  may  follow  that  includes 
the  note  to  which  the  raised  fifth  ascends. 

The  fifth  of  the  Dominant  chord  in  Major  keys  may  be  raised  when 
the  seventh  is  added;  in  general  the  raised  fifth  is  put  above  the 
seventh. 


94 


HARMONY. 


=5" 


i 
i 


1 .  C  with  augmented  fifth,  root  at  bass. 

2.  C  with  augmented  fifth,  third  at  bass. 

3.  C  with  augmented  fifth,  fifth  at  bass. 

4.  Dominant  seventh,  with  augmented  fifth. 

5.  Succession,  first  and  third  chords,  with  augmented  fifth. 

6.  Dominant  of  C,  augmented  fifth,  third  progression. 

The  augmented  fifth  chord,  produced  by  lowering  the  root  of  a 
minor  chord,  is  always  followed  by  either  the  second  inversion  of  the 
tonic  (Example  1  ),  or  by  the  dominant  (Example  2 )  of  its  relative 
major. 

It  is  never  inverted,  being  quite  ineffective  except  with  the  root  at 
the  bass. 

1.  2. 


— ^ — 

~3 — H 

— ^  ^  

-  * 

f  ¥ 

»~r  -  -  r 

What  is  known  as  the  chord  of  Passing  Seventh  may  be  conve- 
niently included  in  this  group  of  chords  with  passing  tones. 

The  passing  seventh  may  be  added  to  any  chord  ;  it  may  be  either 
Major  or  Minor  ;  it  always  descends. 

11  J 


1=4= 


I— I—1 


I 


1.  Tonic,  with  passing  seventh. 

2.  Subdominant,  with  passing  seventh. 


HARMONY. 


95 


3.  Mediant,  with  passing  seventh. 

4.  Submediant,  with  passing  seventh.  In  this  case  the  seventh  is 
at  the  bass,  as  the  chord  may  be  inverted  just  as  a  dominant  seventh 
is.  In  fact,  it  will  be  seen  that  it  progresses  exactly  like  a  dominant 
seventh  ;  thus,  first,  third,  fourth,  fifth  resemble  dominant,  followed 
by  tonic;  2  is  like  a  dominant  with  second  progression. 

5.  Tonic  with  passing  seventh,  with  fifth  at  bass. 

When  the  passing  seventh  is  Major,  which  is  only  so  with  the 
Tonic  and  Subdominant  chords,  the  augmented  fifth  is  often  used 
with  it. 


I.  x 

2-  x 

-X  \l  A. 

— •  

X 

m  *-     ?  • 

The  use  of  the  Aug 

•rnented  fifth  fu 

rnishes  a 

new  way  of  treating 

raised  notes.  The  only  one  that  may  not  be  so  treated,  without 
going  outside  the  Related  Group,  is  the  raised  fourth.  It  enables 
us  to  raise  the  sixth  of  the  scale,  thus  : 


First. 


Second. 


Fifth. 


Sixth. 


-1 


r4- 

-f- 


J — X 


=1 

(5- — 


-3- 


fc=i 


I 


Questions  on  Chapter  XX.  (Continued.) 

How  may  the  chord  of  the  augmented  fifth  be  produced? 

How  does  the  raised  fifth  move? 

What  chords  may  follow  the  augmented  fifth? 

May  the  fifth  be  augmented  in  the  dominant  seventh  chord?    In  both  major 

and  minor  keys  ? 
How  is  this  chord  generally  arranged  ? 

By  what  chords  may  the  augmented  fifth  chord,  produced  by  lowering  the 
root  of  a  minor  chord,  be  followed  ? 


HARMONY. 


May  this  augmented  fifth  chord  be  inverted  ?    May  the  former  one? 
To  what  chords  may  the  passing  seventh  be  added  ? 
How  does  it  move  ? 
Is  it  major  or  minor  ? 

May  a  chord  with  passing  seventh  be  inverted? 
What  chord  does  it  resemble  in  its  progressions? 
In  which  chords  is  the  passing  seventh  major? 
What  may  be  used  with  it  in  these  cases? 

Which  raised  notes  in  the  scale  may  be  treated  as  augmented  fifths  ? 
What  new  raised  note  may  be  introduced  as  an  augmented  fifth? 


Exercises  for  Augmented  Fifth  and  Passing  Seventh. 


s — 

— 

£2 

u 



23: 


i  

^  1  

-HI 

III. 


2=t 


Z2I 


IV. 


HARMONY. 


97 


CHAPTER  XXI. 

Suspensions. 

( Note.  Suspension  and  Retardation  are  treated  in  a  new  way  in  this  work,  the  intention 
being  to  bring  out  clearly  the  difference  between  them.) 

The  seventh  and  ninth,  Major  or  Minor,  may  be  added  to  any  com- 
mon chord  by  Suspension;  i.  e.,  by  tying  the  note  that  becomes  the 
seventh  or  ninth  from  a  preceding  chord  in  which  it  is  a  consonant 
member  (  root,  third,  or  fifth  )  . 

The  Suspended  seventh  and  ninth  resolve  by  descending  one  degree. 
The  remaining  notes  follow  the  rules  given  for  the  progression  of 
the  dominant  seventh  and  ninth  {<q.  v.). 

The  rule  may  be  put  thus :  the  root  of  the  chord  in  which  the  sus- 
pended seventh  is  resolved  is  always  the  fourth  above,  second  above, 
or  third  below  that  of  the  chord  in  which  the  suspension  occurs. 


1.  Chord  of  with  suspended  seventh  (  D )  ;  root  ascends  a 
fourth  (  like  first  progression  of  dominant  seventh)  . 

2.  Same;  root  ascends  a  second  (like  second  progression  of 
dominant  seventh  ) . 

3.  Root  (C)  is  third  below  (like  third  progression  of  dominant 
seventh) . 

Diatonic  sequences  may  be  made,  in  which  every  chord  has  a 
suspended  seventh. 


Any  Diatonic  Sequence  may  be  turned  into  an  harmonic  sequence- 
The  above  may  be  changed  to  the  succession  of  dominant  sevenths 
(  by  lowered  thirds) . 


II 


i  •—  



n  i- 

-  f 

■  i 

The  suspended  seventh  may  be  inverted  like  the  dominant  seventh. 


The  suspension  belongs  more  to  the  old  diatonic  or  contrapuntal 
system  than  to  the  harmonic. 


HARMONY. 


99 


CHAPTER  XXII. 

Retardation. 

When  a  note  is  prolonged  from  a  chord  in  which  it  is  a  member,  into 
one  in  which  it  is  not  a  member,  thus  becoming  dissonant,  it  is  called  a 
Retardation  of  the  member  of  the  second  chord  upon  which  it  resolves. 

The  Retardation  differs  from  the  Suspension  in  the  following 
respects : 

First,  there  is  no  change  in  the  harmony  when  it  resolves. 

Second,  the  Suspended  seventh  and  ninth  are  always  found  in  con- 
junction with  a  common  chord,  while  the  retardation  may  take  place 
in  any  group. 

Third,  the  Suspension  is  always  prepared  by  a  consonant  member 
of  some  chord,  viz.,  root,  third,  or  fifth,  while  a  Retardation  may  be 
made  by  prolonging  any  member  of  a  chord,  consonant  or  dissonant. 

Fourth,  Suspensions  always  resolve  by  Descending,  while  Retar- 
dations may  either  Ascend  or  Descend. 

The  member  of  the  first  chord  is  the  Preparation ;  the  tied  note 
(or  repeated  note)  is  the  Retardation. 

The  member  of  the  second  chord  (that  follows  the  tied  note)  is  the  Reso- 
lution or  Retarded  note. 

When  a  Retardation  resolves  by  descendifig,  its  descent  may  be  a 
whole  or  half-tone.  When  it  resolves  by  ascending  it  must  be  a  half- 
tone when  the  note  to  which  it  ascends  is  the  root  or  fifth  of  the 
chord;  if  it  ascends  to  the  third  it 'may  be  a  whole  or  half-tone. 

The  note  upon  which  a  Retardation  resolves  may  be  repeated  in 
a  lower  part,  but  not  in  a  part  above  the  Retardation.  (Frequent  excep- 
tions to  this  last  rule  may  be  found,  but  they  rarely,  if  ever,  sound  well.) 


IOO 


HARMONY. 


12. 


(tt  ia  (2  ^  %  -(22  -  —  ~  *  


J  -J- 


2± 


mm 


3: 


-(22- 


-f=<- 


I,  2,  3.  Retardation  of  the  root  from  above;  the  second  chord  has 
root,  then  third,  then  fifth  at  the  bass. 

4,  5,  6.  Retardation  of  root  from  below,  etc. 
7,  8,  9.  Retardation  of  third  from  above. 


10,  II,  12, 

13'  H»  I5- 


Retardation  of  third  from  below. 
Retardation  of  fifth  from  above. 


16.  Retardation  of  fifth  from  below. 

This  Retardation  may  take  place  only  in  this  progression  ;  viz.,  from  supertonic  harmony  or 
augmented  sixth  to  tonic. 

17.  Seventh  retarded  from  below. 

18.  Retardation  of  seventh  from  above.  In  this  the  root  (  G)  is 
above  the  ninth  {A).  This  is  possible  only  when  it  is  used  as  a 
retardation  of  the  seventh. 

Two  or  three  members  of  a  chord  may  be  retarded  at  the  same 
time. 

1.  2.  3.  4.       ,       5.  6. 


r       r       i       1       -— '    itt  " 


pE3 


f 

m 


HARMONY. 


101 


^ "f^^S"""""-*'  g    Tl  R°°t  anQl   third,   retardation  from 


— F-^S-f  H  above. 


2.  Root  below,  third  above. 

3.  Root  and  third  from  below 
zj            —       U       4.  Root  and  third  from  above. 


5.  Root,  third,  and  fifth  from  above. 

6.  Root  from  below,  third  and  fifth  from  above. 

7.  Root  from  above  and  below,  third  and  fifth  from  above. 

The  next  example  gives  the  retardation  in  the  bass,  first  of  the  third 
from  above.  This  is  the  best,  because  it  is  not  necessary  to  have  the 
retarded  note  repeated  in  an  upper  voice,  as  is  the  case  with  2  and  3. 


§ 


9±= 

-  -i 

A 

 1 

-A— —A-  J  

l  r 

—6*  



— # — 

4 

— &  

. — * — 

2  and  3  belong  more  properly  to  that  kind  of  syncopation  in  which 
one  or  more  parts  drag  behind  the  rest,  as  follows  : 


t 


e— g 


— r  r-r     *  ' 


— si- 


The  bass  may  move  to 
another  member  of  the  chord 
at  the  same  time  the  retarda- 
tion resolves,  or  before  it. 


i 


4SL 


T   T  I 


1 


1 


102 


HARMONY. 


In  the  examples  which  follow  will  be  found  a  series  of  sevenths 
treated  in  three  ways,  which  will  serve  to  make  the  difference  be- 
tween Suspension  and  Retardation  clear. 

1  is  a  sequence  which  may  be  considered  as  a  series  of  first  inver- 
sions with  Retardation  of  the  roots. 

2  is  a  sequence  of  real  Suspended  Sevenths. 

3  is  so  harmonized  that  the  seventh  over  the  bass  is  in  turn  the 
Retardation  of  every  member  of  a  chord  from  root  to  ninth. 


In  3,  at  a,  the  root  (  G  )  becomes  a  retardation  of  the  seventh  ( J?), 
because  the  ninth  (  Ah )  is  below  it.  This  is  the  only  way  in  which 
the  ninth  may  be  sounded  below  the  root.  b.  The  third  is  retarded. 
At  c  the  fifth ;  at  d  the  root ;  at  e  the  third ;  and  at  f  the  ninth 
(  A  )  is   retarded  by  the   prolongation  of  B  ( the   third  )  over  C 


HARMONY.  103 

(the  eleventh).  This  is  the  only  way  in  which  the  third  and 
eleventh  may  be  sounded  together  (  with  the  third  above  ) . 


Questions  on  Chapter  XXII. 
What  is  a  retardation? 

In  what  respect  does  it  differ  from  a  suspension  ? 
What  accompanies  a  suspension  ?    What  a  retardation  ? 
How  is  a  suspension  prepared  ?    How  a  retardation  ? 
How  does  a  suspension  resolve  ?    How  a  retardation  ? 
When  a  retardation  descends,  what  may  the  interval  be? 
When  it  ascends? 

When  may  the  note  of  resolution  be  repeated  ? 
May  double  retardations  occur?  Triple? 
What  retardations  are  possible  at  the  bass? 
What  movement  may  the  bass  make  ? 


104 


HARMONY. 


IV. 


^=1 


:t: 


 1  hi — j — j — f—ff — «— * 

II 

V. 


-|  1- 


"i — r^f 


Rewrite  the  following  succession  of  chords  with  as  many  retarda- 
tions as  possible. 


Proceed  as  follows : 


Suspensions  and  Retardations  do  not  always  move  directly  to  their 
Resolution,  but  postpone  it  by  the  introduction  of  certain  other  notes. 
This  is  called  Ornamentation  of  the  Dissonances.  A  Suspended 
seventh  or  ninth  may  ascend  one  degree,  then  fall  a  third  to  the  reso- 


HARMONY. 


lution  (Example  i)  ;  or  may  fall  a  third,  then  ascend  one  degree  to 
the  resolution  (Example  2). 

In  both  cases  the  descent  of  a  third  may  be  filled  by  adding  the 
intermediary  note  (Examples  3  and  4). 


7799 


__i — , — 1 — .  1 — -J^^-^^—m  

 g 

-(2  h— 

zj 

_ 

Cf     J-  ' 

T  F 

4— 

The  same  ornamentation  may  be  g'iven  to  a  descending  Retarda- 
tion (  Example  1  ) . 

An  ascending  Retardation  may  leap  to  the  note  above,  then  fall  to 
its  resolution  (Example  2)  ;  or  the  ascent  of  a  third  may  be  filled  as 
in  the  preceding  cases.  Lastly,  an  ascending  or  descending  retarda- 
tion may  leap  to  any  member  of  the  chord  in  which  it  occurs,  before 
going  to  its  resolution  (Example  3)  ;  and  the  interval  from  the 
retardation  to  this  note,  or  from  this  note  to  the  resolution,  may  be 
changed  to  a  run  (Example  4.) 


io6 


HARMONY. 


=3==t*=l=z 


— I — 1-— I — I- 


-t— 


-I  J  r.^u.jjJU-. 


=\— 

 i 

—  <Sf  

o — 

1 


The  ornamentation  may  be  varied  in  many  ways,  but  the  underly- 
ing principle  is  always  the  same;  viz.,  a  movement  from  one  side  to 
the  other  of  the  Resolution  note,  upon  which  the  dissonance  finally 
comes  to  rest,  varied  by  a  leap  or  run  to  some  other  member  of  the 
chord. 

J — , — J — U^-l — i — i — i — 1 — u=2:-i — t — , — i — ■ 


—  — 

-ITS   lAHTli  J-;£3-£  4- 

9*  z?        p  9 

'   " 

1 .  Retardation  falls  to  another  note  of  the  chord,  then  leaps  back 
to  the  dissonance. 

2.  Same,  but  runs  up  to  note  below  resolution,  then  leaps  to  note 
above  it. 

3.  Descends  to  note  below  resoluLion. 

4.  Runs  to  note  above  resolution. 

5.  There  are  two  chords  in  this  bar  :  first.       A,  C,  is  implied ;  .the 


HARMONY. 


107 


E  is  at  first  a  retardation  of  F,  then  of  D  in  chord  D,  JR.  A.  From 
this  it  may  be  seen  that  the  passing'  seventh  (Chapter  XIX.),  al- 
though itself  a  dissonance,  may  be  used  to  prepare  another  disso- 
nance. 

6.  Leap  to  another  member  of  chord,  then  to  note  below  and  note 
above  resolution. 

7.  Leap  to  note  below,  then  to  another  member  of  chord,  from 
which  it  runs  down  to  the  resolution. 

Double  Retardations  may  be  ornamented,  provided  they  move  to- 
gether in  thirds  or  sixths,  or  provided  the  same  variety  of  ornamen- 
tation is  applied  to  both. 


I 


?5 


11 


Questions  on  Chapter  XXII.  (Continued.) 

Do  suspensions  and  retardations  always  move  directly  to  their  resolution? 
What  is  this  introduction  of  other  notes  called  ? 

Describe  the  various  kinds  of  ornamentation  that  both  suspensions  and  retar- 
dations may  have. 
May  the  passing  seventh  be  used  to  prepare  a  retardation? 
May  double  retardations  be  ornamented? 


Exercises  on  Ornamented  Suspensions  and  Retardations. 


— — - 

-J-J-J-P- 

m 

• 

M — 

"-II 

io8 
II. 


HARMONY. 


in. 


era 


IS 


The  augmented  fifth  may  be  used  as  a  preparation  of  a  retar- 
dation. 


IV. 


=1 

s^'    ■     -1  ■      L  f  -        1  ■ 

t3= 

+^4 

HARMONY. 


109 


CHAPTER  XXIII. 

Changing  Notes  or  Appoggiaturas. 

These  differ  from  Retardations  only  in  being  Struck  instead  of 
being  Prolonged  from  a  preceding  chord.  They  fall  on  the  accentea 
beats,  or  if  there  are  two  notes  or  more  to  the  beat,  on  the  first  of 
each  beat. 

They  may  fall  on  an  unaccented  beat  if  the  accompanying  har- 
mony is  syncopated. 

When  below  the  member  of  the  chord  upon  which  they  resolve, 
they  must  be  a  half-tone  below  it,  except  when  they  occur  as  part  of 
an  ascending  diatonic  scale. 

All  accidentally  raised  notes  that  fall  on  the  accented  beats,  or  on 
the  first  note  of  a  beat,  may  be  treated  as  Changing  Notes,  when  they 
ascend  one  degree. 

They  may  be  doubled  in  thirds  and  sixths  like  Retardations. 


1.  On  accented  beats. 

2.  On  first  note  of  beat. 


r  10 


HARMONY. 


3.  On  second  beat  (unaccented),  the  harmony  syncopated. 

4.  The  D,  a  whole-tone  below  E.  Like  the  retardation,  the 
changing  note  may  be  a  whole-tone  below  the  third  of  a  chord. 

5.  Raised  notes  on  first  of  beat. 

6.  Doubled  in  thirds  and  sixths. 

7.  Same,  with  raised  notes. 

A  Dominant  harmony  may  be  struck  over  the  root  or  third  of  the 
tonic,  making  Double,  Triple,  or  Quadruple  Changing  Notes.  The 
dominant  harmony  moves  according  to  the  rule  (1)  and  (2). 

A  changing  note  below  the  third  and  one  above  the  fifth  of  a  com- 
mon chord  may  be  struck  over  the  root,  or  root  and  fifth  (3)  and  (4.) 


The  Changing  Note  may  be  ornamented  like  the  Retardation. 


HARMONY. 


in 


Questions  on  Changing  Notes. 

1.  In  what  respect  do  changing  notes  differ  from  retardations? 

2.  On  which  beat  of  the  measure  do  they  fall? 

3.  May  they  occur  in  any  other  situation  ? 

4.  When  do  they  fall  on  an  unaccented  beat? 

5.  When  may  they  be  a  whole-tone  below  the  note  of  resolution? 

6.  When  may  accidentally  raised  notes  be  treated  as  changing  notes? 

7.  May  changing  notes  be  doubled  ? 

Exercises  on  Changing  Notes. 

The  note  upon  which  the  Changing  Note  resolves  may  not  be  struck 
with  it,  but  it  may  be  struck  at  the  octave  below. 
I. 


H=f= 


-t-0-  


III. 


Ornamented  Changing  Notes. 


  ft* 

__j  _l    m.  rrq 

• — ^ — 

0— 

wir  ~  t=t  it: 

=3 

0 

1 — A — * 

112 


HARMONY. 


CHAPTER  XXIV. 

Passing  Notes. 

The  last  manner  in  which  Dissonant  unharmonized  notes  may  be 
used  is,  as  Passing  Notes. 

Passing  Notes  occur  between  the  harmonies  on  the  unaccented 
beats,  or  on  the  second  note  of  the  beat. 

There  are  Five  Varieties. 

The  First  Variety  enters  by  degrees  forming  part  of  an  ascending 
or  descending  diatonic  or  chromatic  scale.  To  get  a  passing  note 
of  this  variety  there  must  be  at  least  three  ascending  or  descending 
notes  in  succession,  the  middle  one  of  which  may  be  a  passing  note. 


5EE 


=t 


a 


a.  The  A  must  ascend  and  double  the  third  (  B  )  in  the  next  chord, 
to  avoid  making  the  parallel  fifths,  A,        G,  D. 

b.  The  third  (-£•)  is  doubled  here,  because  if  the  root  (C)  were 
doubled  it  would  clash  with  the  passing  note  ( D  ) . 

c.  This  is  the  only  way  in  which  a  note  may  be  natural  and  sharp 

at  the  same  time.  (Theoretically  this  should  be  D\l,  but  for  all  practical  purposes  it 
is -more  convenient  to  treat  it  as  a  chromatic  passing  note.) 

Diatonic  or  chromatic  scales  of  any  length  may  be  played  with  a 
single  chord,  provided  the  first  and  last,  or  second  and  last  notes 
are  members  of  the  chord.  If  the  second  and  last  are  members  of  the 
chord,  the  first  note  is  a  Changing  Note.  In  passages  of  this  kind 
it  is  convenient  to  consider  all  the  intervening  notes  as  passing  notes. 
The  chord  may  be  changed  with  the  last  note. 


HARMONY. 


"3 


First  and  last. 


Second  and  last.      First  and  last. 


Second  and  last. 


The  direction  of  the  scale  may  be  changed  whenever  a  note  belong- 
ing to  the  chord  is  reached,  or  the  scale  may  be  abandoned  when  such 
a  note  is  reached. 


0., 


<r.  d. 

X  X  XX 


a.  In  the  first  measure  each  time  a  member  of  the  chord  is 
reached,  the  motion  is  changed,  the  next  group  of  sixteenths  begin- 
ning with  a  changing  note. 

b.  The  change  of  direction  begins  with  the  member  of  the  chord. 

c.  Is  a  passing  note  of  the  second  variety. 

d.  The  scale  is  abandoned.  There  is  a  leap  from  one  member  to 
another  of  the  chord. 

The  Second  Variety  enters  by  degrees  above  or  below  the  harmon- 
ized note,  but  returns  to  the  same  note.  All  varieties  of  turn,  trill, 
mordent  are  founded  on  this  variety. 

Like  the  Retardations  this  variety  of  passing  note,  when  below  the 
harmonized  note,  sounds  better  if  a  half-tone ;  but  it  may  be  a  whole- 
tone  below  the  third  of  a  chord. 


9 


i 


g:  .g. 


HARMONY. 


The  Third  Variety  enters  by  a  leap  from  one  harmonized  note  to 
the  degree  above  or  below  the  next  harmonized  note. 

When  below  it  must  be  a  half-tone.  All  accidentally  raised  notes 
that  occur  on  the  unaccented  beats,  or  the  second  member  of  the 
beat,  may  be  treated  as  passing  notes  of  this  variety. 


x  x 


4 


The  Fourth  Variety  may  be  described  as  an  Ornamented  Diatonic 
scale,  made  by  adding  to  each  note  of  the  scale  the  note  above  or 
below,  or  a  third  above  or  below.  So  many  variations  may  be  made 
in  this  variety  that  it  is  only  possible  to  give  a  few  examples. 


1 


-g>- 


^2- 


I 


3 


I 


i  is  founded  on 
this  diatonic 


< — *5  =  

*  •  a 

P  j  i 

< 

■i  n* 

6  on 


HARMONY. 


Some  apparently  elaborate  passages  may  be  reduced  to  very  simple 
harmonic  elements  by  eliminating  the  passing  notes;  for  example, 
the  following  from  Handel. 


I 


— « — = 

X 

l*»r —  ■ 

r —  i! 

The  Fifth  Variety  is  called 
the  Anticipating  Note.  It  just 
reverses  the  Retardation  ;  i.e., 
it  anticipates  the  following 
chord  by  sounding  one  of  its 
members  before  the  preceding 
chord  is  left. 


X   X  X 


In  the  example  that  follows,  all  these  varieties  are  used  in  a  cadenza- 
like passage  over  a  single  chord. 

x  x 


Observe  that  every  change  in  the  direction  of  the  motion  begins 
with  a  member  of  the  chord. 


n6 


HARMONY. 


All  the  ornamentation  that  may  be  given  to  the  retardation  and  the 
changing  note  may  be  given  to  the  first  three  varieties  of  the  passing 
note. 

i.  2.  3.  4.  5.  6. 


0- 

• 

•  

P 

— — 

•  — 

1 

# 

w- 

9— 

 « 

— 



* 


i 


1.  First  variety,  leaping  to  note  below. 

2.  To  another  member  of  the  chord. 

3.  Second  variety,  leaping  to  note  below. 

4.  To  another  member  of  the  chord. 

5.  Third  variety,  leaping  to  note  above. 

6.  Leaping  to  note  below. 

The  first  three  varieties  may  be  doubled  in  thirds  and  sixths  (1). 
The  first  (  diatonic  )  may  also  be  doubled  at  the  octave  or  third, 
when  moving  in  opposite  directions  (2). 

The  second  and  third  varieties  may  be  doubled  at  augmented 

fourth  or  diminished  fifth  (like  changing  notes)  (3). 

A  diatonic  scale  of  first  inversions  may  be  written  over  a  single 
note  or  chord  (4) . 

Diatonic  or  chromatic  scales  may  be  written  in  contrary  motion, 
in  either  single  notes,  thirds,  or  sixths,  provided  the  first  and  last 
notes  struck  together  belong  to  the  same  harmony  (5). 

A  passing  note  is  sometimes  followed  by  a  changing  note  (6). 

The  resolution  of  a  passing  note  is  sometimes  retarded  by  repeat- 
ing it  (7)  ;  or  by  leaping  to  some  other  member  of  the  chord  and 
returning  to  the  passing  note  (8). 

First.       ^  Second.  Third.  ^ 


9i 


HARMONY. 


Third  Sonata.  Beethoven. 

8va 


Examples  maj  be  found  of  changing  and  passing  notes  used  in  a 


manner  that  does  not  conform  to  any  of  the  rules  here  given,  but 
they  are  rare,  and  will  cause  the  pupil  no  difficulty  if  these  rules  are 
well  understood. 

Questions  on  Passing  Notes. 
What  is  a  passing  note? 

How  many  varieties  are  there  of  passing  notes? 
Describe  the  first  variety. 

How  many  notes  must  there  be  in  succession  to  have  a  passing  note  of  this 
variety  ? 

How  may  extended  diatonic  and  chromatic  scales  be  treated  ? 
Describe  the  second  variety  of  passing  note. 
Describe  the  third  variety? 

When  may  accidentally  raised  notes  be  treated  as  passing  notes? 
Describe  the  fourth  variety. 
Describe  the  fifth  variety. 

May  the  ornamentation  used  with  retardations  and  changing  notes  be  applied 

to  passing  notes? 
To  all  varieties?  .  . 

In  what  ways  may  passing  notes  be  doubled  ? 


Exercises  on  Passing  Notes  and  Changing  Notes. 


HARMONY. 


119 


CHAPTER  XXV. 
Further  Remarks  on  the  Minor  Scale. 

(it  was  necessary  to  defer  this  chapter  until  changing  and  passing  notes  were  explained.) 

The  changeable  notes  in  the  minor  scale  offer  some  difficulties  to 
the  student.    The  following  instructions  will  make  this  point  clear. 

In  scale  passages  ascending,  either  the  harmonic  or  melodic  form 
may  be  used  when  accompanied  by  the  tonic  chord  (1)  . 

When  accompanied  by  the  subdominant  chord,  the  harmonic  or 
natural  form  may  be  used  (2). 

When  accompanied  by  the  dominant,  root  to  seventh,  the  melodic 
or  harmonic  form  ;  but  if  the  dominant  harmony  is  third  to  ninth, 
or  jifth  to  eleventh,  the  harmonic  form  must  be  used  (3). 

Descending  scale,  tonic  chord  accompanying,  use  natural  or  har- 
monic (4). 

Subdominant  accompanying,  use  natural  form  (5). 
Dominant  accompanying,  use  harmonic  form  (6). 
The  form  given  in  7  is  much  used  by  Bach  and  Handel.    It  dif- 
fers from  the  major  scale  only  in  having  a  minor  third  above  the  tonic. 


120 


HARMONY. 


The  raised  sixth  of  the  minor  scale,  when  in  the  melody,  may  be 
harmonized  as  in  the  example  following,  i  ;  when  in  the  bass,  2. 
These  both  belong  to  the  old  school  of  music.  Some  writers  say  that 
2  should  never  be  used,  but  3,  from  the  ''Duetto"  (Songs  Without 
Words),  is  a  sufficient  argument  for  its  use.  4  and  5  give  the  usual 
ways  of  harmonizing  it.  6.  It  is  treated  as  a  passing  note  on  the 
tonic  chord.  *j.  It  is  treated  as  a  changing  note  in  the  dominant 
chord. 


The  natural  seventh  descending:  1.  Diatonic  harmony,  rugged 
but  effective  on  occasion.  2.  The  same  passage  in  the  bass.  This 
manner  of  using  the  first  inversion  of  the  minor  chord  on  the  domi- 
nant, viz.,  when  the  bass  descends  diatonically  from  the  keynote  to 
the  sixth  or  fifth  of  the  scale,  is  very  effective.     Some  authors  say  it 


HARMONY. 


121 


is  the  only  way  this  chord  may  be  used,  in  spite  of  the  fact  that 
self-willed  composers  often  use  it  otherwise  with  excellent  effect. 
3  *nd  4  give  modern  ways  of  harmonizing  this  note.  5  and  6 
give  examples  of  the  use  of  the  raised  sixth  in  descending  passages; 
in  5  the  sixth  returns  to  the  seventh.  In  6  the  sixth  and  seventh  are 
both  raised,  although  in  a  descending  passage ;  the  seventh  because 
the  harmony  requires  it,  the  sixth  to  avoid  the  skip  of  augmented 
second,  which  is  very  disagreeable  in  the  bass. 


cond,  which  is  very  disagreeable  in  the  bass. 


The  augmented  fifth  on  the  third  degree  of  the  minor  scale  may 
be  used.    It  is  generally  used  as  a  retardation,  (1). 

2  shows  how  the  natural  seventh  may  be  used  as  a  passing  or 
changing  note  with  the  raised  seventh. 


2. 

(i  *  % 

ik^  -i 

 %  

 0 

=1 

2. 

?  11 

 J  

li 

 1  .  

Exercises  in  Minor 

I. 

Keys. 

1  I 

 1  1— 

m 

^  1 

HARMONY. 


123 


CHAPTER  XXVI. 


Open  or  Vocal  Harmony. 


The  way  in  which  the  exercises  have  been  written  so  far  is  called 
Close  Harmony  ;  i.  e.,  the  three  upper  parts  are  kept  as  close 
together  as  possible. 

This  has  been  done  because  it  is  the  easiest  way  in  which  chord 
successions  may  be  written,  and  because  of  a  belief  that  after  the 
pupil  has  learned  the  chords  and  the  rules  that  govern  their  motions, 
the  writing  of  them  in  Open  or  Vocal  score  is  very  much  easier  than 
when  the  pupil  has  not  only  the  rules  for  open  harmony,  but  also  for 
the  formation  and  succession  of  the  chords  to  remember  at  the  same 
time. 

Any  correctly  written  example  of  chords  in  Position  may  be 
turned  into  Open  Harmony  by  the  simple  expedient  of  moving  the 
middle  note  of  the  upper  three  down  an  octave,  thus  : 

a. 


This  expedient  fails  when  Inversions  and  Dissonant  chords  are 
used  ;  for  example,  the  following  is  right  in  close  harmony,  but  when 
written  in  open,  three  successive  fifths  will  be  found  at  <z,  and  a  poor 
arrangement  of  the  voices  at  3,  the  tenor  and  bass  being  too  close, 
and  the  soprano  and  alto  too  far  apart. 


124 


HARMONY. 


Rules.  All  previous  rules  as  to  Progression  and  Resolution  must 
be  strictly  observed. 

No  two  parts  must  ever  move  together  in  Octaves,  Fifths,  or 
Unisons. 

Avoid  the  following  leaps:  augmented  second,  fifth,  sixth; 
diminished  third,  fourth  ;  major  seventh,  minor  seventh  ;  except  when 
from  root  to  seventh  of  dominant  chord,  all  leaps  beyond  the  octave. 

Avoid  all  long  leaps  in  the  Inner  parts  (Alto  and  Tenor).  Never 
allow  the  Alto  and  Tenor  to  be  separated  by  an  interval  greater  than 
an  octave. 


Soprano. 


Alto. 


Keep  the  voices  within  the 
following  limits,  using  the 
highest  and  lowest  notes  of 
each  voice  rarely. 


Tenor. 


Bass. 


9± 


2? 


This  kind  of  writing  should  also  be  practised  on  four  staves  (one 
for  each  voice) .  It  is  customary  now  to  use  the  G  clef  for  the  Tenor, 
with  the  understanding  that  its  pitch  is  an  Octave  Lower  than  when 
used  for  the  Soprano. 

The  example  (marked  A),  page  124,  is  here  given  on  four  staves. 

.  Soprano. 


0 


Alto 


Rewrite  all,  or  as  many  as  possible  of  the  preceding  exercises, 
except  those  on  changing  and  passing  notes,  in  open  harmony,  after 
studying  the  following  examples.  This  department  belongs  more 
especially  to  the  study  of  counterpoint. 


HARMONY. 


Observe  that  when  a  note  belongs  to  two  or  more  chords  in  suc- 
cession, it  may  be  written  as  a  single  note  equalling  the  others  in 
value ;  as  in  second  measure,  first  example. 

When  a  letter  is  altered  by  a  sharp,  flat,  or  natural,  keep  it  in  the 
same  voice. 


— <5<-q  


I        1^  I 


(2— 


m 


9?l 


is?— 


Mil 


n 


— ,-•—-! — \-x~r^— n 


f2-     -<5<-     -  -<5>- 


* .  • 


Ffff 


126 


HARMONY. 


CHAPTER  XXVII. 


Pedal  or  Organ  Point. 


By  this  is  meant  the  prolongation  of  a  single  sound  through  several 
measures,  generally  in  the  lowest  part,  while  a  succession  of  harmo- 
nies related  to  the  key  is  written  over  it. 

These  harmonies  must  be  written  as  though  the  Pedal  Note  were 
not  present.  But  the  Pedal  Note  must  be  a  member  of  at  least  the 
first  and  last  chords. 

The  Dominant  is  most  frequently  used  as  a  pedal ;  next,  the 
Tonic ;  next,  the  Tonic  and  Dominant  together. 

Examples  of  other '  degrees  of  the  scale  used  as  pedals  are  rare. 
Examples  may  be  found  of  tonic,  dominant,  and  the  fifth  of  the 
dominant  (i.  e.,  the  supertonic)  used  together. 

Progressions  may  be  written  with  the  greatest  freedom  over  the 
Dominant;  next,  over  the  Tonic. 

They  are  necessarily  much  restricted  when  both  these  notes  are 
used,  and  still  more  when  the  three  notes  mentioned  above  are  used. 

If  the  prolonged  note  is  a  member  of  all  the  chords  written  over  it, 
viz.,  root  or  fifth,  it  is  not  strictly  speaking  a  Pedal.  This  kind  of  pro- 
longed note  is  often  used  in  an  upper  part.  (The  real  Pedal  is  rarely  so  used, 
and  the  harmonies  that  may  be  written  with  it  are  much  restricted.) 

In  Pianoforte  and  Orchestral  music  the  Pedal  note  is  often  repeated 
or  alternated  with  its  octave,  or  with  the  half-tone  below  it. 

A  succession  consisting  of  the  Tonic,  Dominant,  and  Tonic  is 
often  written  over  the  Tonic,  which  might  then  be  termed  a  Pedal  of 
short  duration. 


■  ¥ 


HARMONY. 


128 


HARMONY. 


1.  Dominant  Pedal. 

2.  Tonic  Pedal. 

3.  Tonic  and  Dominant  together. 

4.  Tonic,  Dominant,  and  Supertonic.  This  is  rarely  used  unless 
a  quasi  rustic  effect  is  desired.  It  must  end  by  the  supertonic  falling 
to  the  octave  of  the  tonic. 

5.  Example  of  a  prolonged  note  that  is  a  member  of  every  chord, 
therefore  not  strictly  a  pedal. 


•)■  sS  jS 


— 1  41  1 


-0— i  1  1  L-H  M  1 — 1 — h — # — £  \-vA  \ 


I.  The  Pedal  repeated,  alternated  with  its  octave  and  with  the 
half-tone  below. 


HARMONY. 


129 


2.  Pedal  on  the  tonic,  of  short  duration. 

3.  Is  a  beautiful  illustration  of  this  variety  of  pedal,  from  Chopin's 
Cradle  Song.  This  accompaniment  persists  all  through  the  piece. 
It  is  simply  a  breaking  or  dispersion  of  the  following  chords. 

I:  i- 
g@  II 

Pedals  of  considerable  length  may  be  met  with  in  which  transitions 
are  made  to  non-related  keys,  but  as  a  rule  the  harmonies  are  con- 
fined to  the  Related  Group. 


13°  HARMONY. 


CHAPTER  XXVIII. 
Transition. 


Transition  is  the  act  of  passing  out  of  the  related  group. 
A  Transition  is  not  established  until  a  Tonic  Chord  foreign  to  the 
Group  is  struck. 

The  presence  of  a  Tonic  chord  may  be  indicated  in  two  ways: 
first,  and  most  emphatically,  by  being  preceded  by  the  harmony  of 
its  own  dominant;  second,  by  appearing  in  its  Second  Inversion, 
provided  the  second  inversion  does  not  enter  in  the  manner  prescribed 
for  a  second  inversion  of  dominant  or  subdominant. 

As  this  second  means  of  making  a  Transition  does  not  necessitate 
the  use  of  Dissonant  chords,  we  will  begin  with  it. 

Taking  first  the  Major  chords  in  the  scale,  each  one  may  be  a  Tonic 
without  transition. 

The  first  and  fourth  may  be  Subdominants  without  Transition; 
but  if  the  fifth  is  treated  as  a  Subdominant,  and  is  followed  by  the 
second  inversion  of  its  Tonic,  it  gives  a  transition,  thus : 


I 


i 


The  chord  G,  B,  D  is  treated 
as  subdominant  of  D,  and  is  fol- 
lowed by  the  second  inversion  of 
D,  which  is  outside  of  the  group. 


The  three  major  chords  may  also  be  dominants ;  the  first  and  fifth 
within  the  group,  the  fourth  gives  a  transition,  thus : 


i  fit :  '  :! 


ii 


5t 


F,  A,  C,  as  dominant,  fol- 
lowed by^b,  D,  F,  out  of  the 

^=zm\\ group- 


HARMONY. 


A  major  chord  is  also  found  on  the  sixth  degree  of  the  minor 
scale.  Of  the  three  major  chords  in  C,  the  first  and  fourth  treated 
thus  remain  within  the  group,  but  the  fifth  passes  out  of  the  group, 
thus  : 


G,  B,  D  is  treated  as  sixth  in  B  minor. 

Transition  by  treating  the  major  chord  as  sixth  in  a  minor  scale 
maybe  extended  by  treating  the  Minor  scale  as  the  parallel  of  a  Major 
scale,  thus : 


S 


1.  C,  E,  G,  as  sixth  of  E  minor,  the  parallel  of  E  major. 

2.  F,  A,  C,  as  sixth  of  A  minor,  the  parallel  of  A  major. 

3.  G,  B,  D,  as  sixth  of  B  minor,  the  parallel  of  B  major. 

A  major  chord  may  also  be  treated  as  being  the  chord  on  the  Low- 
ered Supertonic  of  either  a  Major  or  Minor  scale,  thus  : 


132 


HARMONY. 


1 .  C,  E,  G,  as  lowered  supertonic  of  B,  Major  or  Minor. 

2.  7%  A,  C,  as  lowered  supertonic  of  E,  Major  or  Minor. 

3.  G,  B,  E>.  as  lowered  supertonic  of         Major  or  Minor. 

For  the  sake  of  clearness  the  Tonic  of  the  key  to  which  the  Tran- 
sition is  made  follows  directly  after  the  chord  by  means  of  which  the 
Transition  is  made,  but  as  a  result  of  the  rule  already  given,  that  the 
movement  of  common  chords  is  perfectly  free,  any  of  the  chords  be- 
longing to  the  new  scale  may  appear  before  the  tonic  is  struck. 
For  example,  C,  E,  G,  is  found  in  C,  as  tonic ;  E^  as  dominant, 
G,  as  subdominant.  It  may  therefore  be  followed  by  any  chord  in 
these  keys  or  their  related  keys.  But  when  treated  as  sixth  of  a 
parallel  minor,  or  as  a  lowered  supertonic,  it  is  subject  to  the  rules 
already  given  for  the  use  of  these  chords  (  Chap.  XI.). 

The  three.  Minor  chords  in  the  scale,  treated  as  tonics,  remain 
within  the  related  group. 

As  subdominants,  those  on  the  second  and  sixth  remain  within, 
unless  treated  as  belonging  to  parallel  minor  scales,  thus  : 

„  i-  !     J    I  1     2-  3-ii. 


i 


pi 

m 

-  r  x 

^4 

%  - 

1.  D,  E,  A,  subdominant,  A  minor,  parallel  of  A  major. 

2.  Ay  C,  E,  subdominant,  E  minor,  parallel  of  E  major. 

3.  E,  G,  B,  subdominant,  B  minor;  gives  a  Transition  when 
followed  by  either  B  major  or  minor, 


A  Minor  chord  is  also  found 
on  the  third  of  the  major  scale. 
The  supertonic  is  the  only  one 
that  will  give  a  transition  when 
treated  this  way. 


t 


9± 


II 


In  the  following  example  all  the  transitions  are  made  by  means  of 
common  chords,  treated  in  the  various  ways  just  pointed  out.  The 


HARMONY. 


J33 


places  where  the  transitions  occur  are  marked.  It  is  left  to  the  pupil 
to  explain  them. 


N  l  '  I 
JV!  


^  1- 

 it- — 

J2.- 

• 

-  5= 

■  — %- 

0— 

N.B. 


4* 


«3£ 


r  bf 


N.  B.  The  chords  of  C§  and  D\>  are  enharmonically  the  same. 


to 


zf__at 


9- 


r  r  r 


f  r 


-• — & 


i  i 

Transition  by  means  of  Dominant  Harmonies. 

Any  Dominant  Harmony  may  be  followed  by  either  a  major 


J34 


HARMONY. 


or  minor  tonic.  Thus  the  following  groups  derived  from  G  may 
be  followed  by  C  major  or  C  minor  % 

G     B     D   F—     B     D     F     Ab—£>,     F,     A%  C. 

First.  Third,  Fifth.  Seventh.  Third.   Fifth.  Seventh.  Minor  Fifth.  Seventh.  Minor  Eleventh. 

ninth.  ninth. 

(  The  ninth,  if  present,  must  be  minor,  as  the  major  ninth  cannot  be  used  in  a  minor  key.) 


~£         g*.    or  ~Z7~ 


-J  I- 


u^j  :szo  r  zs: 


2£ 


1 


It  will  at  once  be  seen  that,  by  taking  advantage  of  this  fact,  Three 
Transitions  may  be  made  from  a  given  key  by  changing  the  major 
tonics  to  Minor,  and  three  by  changing  the  Minor  tonics  to  major. 


I 


=t 


afcrifc* 


r 


=^  

 1  

r 


^  J_ij 


=3= 


X 


9 


9 


1.  The  tonic  of  the  principal  key  is  changed  to  Minor,  and  as  C 
minor  is  the  relative  minor  of  Eb,  the  Transition  is  from  C  to  Eb. 

2.  The  same  process  in  the  related  key  of  G  takes  us  to  Bb. 

3.  The  same  process  in  the  related  key  of  F  takes  us  to  Ab. 


HARMONY. 


l35 


This  means  of  making  Transition  may  be  combined  with  that 
first  given.  It  would  require  volumes  of  examples  to  illustrate  them. 
We  give  a  few,  leaving  it  to  the  pupil  to  invent  others, —  an  easy  task 
if  the  principles  are  kept  in  mind. 

To  analyze  a  Transition,  always  observe  what  relation  each  chord 
bears  to  the  chord  and  key  that  precedes  it,  and  to  the  chord  and 
key  that  follows  it. 

r  \x  x  i  x  x 

i.  Dominant  harmony  of  G,  fifth  to  eleventh,  followed  by  G 
minor  (  2  ) .  This  is  then  treated  as  being  a  third  of  iii?,  and  is 
followed  by  dominant  of  E^.  3.  Same  process  repeated  in  E9,  but 
the  minor  chord  (  4  ),  B\>,  is  treated  as  subdominant  of  E  minor,  the 
parallel  of  E  major. 

In  the  next  examples  the  supertonic  harmonics  of  the  three  major 
keys  of  the  group  are  followed  by  minor  tonics. 


HARMONY. 


4* 


zfe 


Write  corresponding  modulations  in  all  keys ;  also  make  the 
corresponding  changes  in  the  related  minor  keys. 

A  Third  means  of  Transition  is  by  Additions  to  common  chords. 

To  a  Major  chord,  the  minor  third  below  its  root  may  be  added ; 
it  then  becomes  fifth,  seventh,  major  ninth,  and  eleventh  of  a  domi- 
nant. This  addition  to  the  first  and  fourth  chords  will  not  give  a 
transition,  unless  the  ninth  is  minor  (  see  former  rule  ) ,  but  it  will 
with  the  fifth. 


m 


1.  C,  E,  G,  with  A,  the  minor 
third  below,  added.  If  the  E 
were  changed  to  E\>,  the  tonic  of 
G  minor  might  follow,  giving  a 
transition  to 

2.  Same  remarks  will  apply. 


3.   G,  B,  D,  becoming  dominant  eleventh  of  D  major. 

The  augmented  sixth  over  the  root  may  be  added  to  any  major 
chord.  This  gives  some  startling  Transitions.  Like  the  supertonic 
harmony,  a  major  or  minor  tonic  may  follow.  (  1  and  2  ),  if  fol- 
lowed by  minor  tonics,  resolve  within  the  group.  (3)?  out  of  the 
group  whether  major  or  minor  tonic  be  used  after  it. 


0 


9i 


w 


According  to  our  rule  the 
raised  note  must  be  the  third. 
Therefore  E$  is  the  root  of  1, 
and  as  F\  is  the  supertonic  of  E, 
this  is  the  augmented  sixth  of  E. 
Root  of  second  is  B ;  of  third,  Cjf. 


HARMONY, 


*37 


A  minor  third  below  the  root  may  be  added  to  a  minor  chord.  It 
may  be  fifth,  seventh,  minor  ninth,  and  eleventh  of  a  dominant,  or 
third,  fifth,  seventh,  and  major  ninth  of  a  dominant  or  supertonic. 

1.  I),  F,  A,  with  B  added,  treated  as  dominant  eleventh  of  A 
major  or  minor. 

2.  Same,  treated  as  dominant  ninth  of  C. 

3.  Same,  treated  as  supertonic  ninth  of  F. 

4  and  5.  Are  the  remaining  minor  chords  in  C,  with  minor  third 
below  added. 

These  and  the  preceding  examples  with  major  chords  should  be 
all  written  out  and  resolved. 

The  addition  of  a  minor  third  above  a  Minor  chord  has  the  same 
result  as  the  addition  of  a  minor  third  below  a  Major  chord. 


x 


The  next  means  of  making  Transition  is  by  the  Chromatic  Altera- 
tion of  chords;  first,  from  major  to  minor,  and  the  reverse;  second, 
changing  a  group  of  four  sounds  into  some  other  group  of  four 
sounds. 

Thus  the  group  G,  B,  D,  F  (first,  third,  fifth,  seventh  of  domi- 
nant of  supertonic)  may  be  changed  into  third,  fifth,  seventh,  ninth, 
by  making  G  sharp. 

Into  fifth,  seventh,  ninth,  eleventh  by  making  B  fiat. 

Into  fifth,  seventh,  ninth,  eleventh,  or  third,  fifth,  seventh,  ninth 
by  making  B  and  D  flat. 

Into  third,  fifth,  seventh,  ninth  by  making  B,  Z>,  and  F  flat. 

Last,  by  flatting  D  it  becomes  an  augmented  sixth  chord,  Z)b,  F, 
G,  and  B. 


i3§ 


HARMONY. 


Write  out  these  changes  and  resolve  them. 


In  the  course  of  this  change 
one  or  more  of  the  sounds  may 
be  enharmonically  changed. 


The  last  means  by  which  Transitions  may  be  made  is  by  Enhar- 
monic Change. 

First,  advantage  may  be  taken  of  the  fact  that  the  dominant  seventh 
chord  and  augmented  sixth  chord  sound  alike,  consequently  the 
seventh  may  be  changed  to  an  augmented  sixth,  or  the  reverse. 

This  change  is  not  always  expressed  in  the  writing. 

n    I.  2.m  u 

:l2§ggj§~r^g-rj      i .  Dominant  of  F,  changed 
~~1       1 1    to  augmented  sixth  of  E. 

i.  Augmented  sixth  of  C, 
 h&  (9 — i  n   changed  to  dominant  of  Lh» 


—& — 


HARMONY. 


At  b  the  augmented  sixth  of  B  is  changed  back  to  the  dominant  of 
C.  The  change  is  implied  here  also,  but  it  differs  from  the  first  case 
in  that  the  chord  is  written  according  to  its  resolution,  whereas  in  the 
first  case  it  is  not. 

The  chief  source  of  enharmonic  modulation  is  found  in  the  dimin- 
ished seventh  chords. 

These  chords  consist  of  four  sounds  separated  from  each  other  by 
three  half-tones ;  thus,   B,     A     F,  A\>. 

Third   Fifth.  Seventh.  Ninth. 

It  will  be  found  that,  no  matter  in  what  order  they  are  written, 
this  will  hold  good  ;  therefore,  any  one  of  the  sounds  may  be  in  turn, 
the  third,  fifth,  seventh,  or  ninth.  Thus  if  A\>  be  put  at  the  bottom 
we  get,  ^4b,  B,  Z>,  F,  and  the  A\>  being  enharmonically  6$,  the  chord 
may  be  written        B,  F. 

Therefore,  as  there  are  four  notes  in  the  chord,  it  may  be  written 
in  four  ways,  and  obtained  from  four  roots,  and  resolved  in  four  ways 
as  a  dominant,  and  four  ways  as  a  supertonic  harmony. 

Take  the  group  B,  D,  F,  A\>  and  move  every  sound  up  a  half- 
tone ;  we  get  C,  Eb,  F§,  A\.  Move  these  up  a  half-tone ;  we  get 
Cjf,  E%  G\  B\>.  Repeat  the  process,  and  we  get  the  group  with 
which  we  began.  It  will  be  seen  that  in  the  three  groups  all  the 
sounds  of  the  Chromatic  Scale  are  included. 


First  group. 


I. 

:f=|= 


Second  group. 

4- 


Third  group. 

3- 

Ww<2-W  4. 

ii-llorbfelM 


First  group,  i.  Written  as  if  derived  from  G ;  therefore  domi- 
nant of  C  major  or  minor,  or  supertonic  of  F  major  or  minor. 

2.  As  if  derived  from  E\  dominant  of  A,  supertonic  of  D. 

3.  As  if  derived  from  Cjjj ;  dominant  of  F$,  supertonic  of  B. 

4.  As  if  derived  from  B\> ' ;  dominant  of  Z?b,  supertonic  of  A\>. 
Second  group.     1.  As  if  derived  from  C;  dominant  of  F,  super- 
tonic of  B\>. 

2.  As  if  derived  from  A ;  dominant  of  Z>,  supertonic  of  G. 


HARMONY. 


3.  As  if  derived  from  F\\  dominant  of  B,  supertonic  of  E. 

4.  As  if  derived  from  D%  or  E\> ';  dominant  of  G#  or  A>,  super- 
tonic  of  C%  or  Z)b. 

Third  group,  1.  As  if  derived  from  Z?;  dominant  of  G,  super- 
tonic  of  C. 

2.  As  if  derived  from  B\  dominant  of  E,  supertonic  of  A. 

3.  As  if  derived  from  G\  or  A\> ;  dominant  of  C$  or  /)!?,  supertonic 
of  F%  or  Gt>. 

4.  As  if  derived  from  F\  dominant  of  B\>,  supertonic  of  E\> '. 

Tt  will  be  seen  that  each  group  is  found  twice  in  every  key;  viz., 
in  the  dominant  of  the  major,  and  in  that  of  its  relative  minor.  This 
being  true  in  the  scale  of  C,  it  follows  that  it  must  be  so  in  every 
scale. 

This  fact  makes  it  possible  to  change  instantly  from  one  key  to 
another  by  means  of  this  chord  in  two  ways. 

First  way,  take  the  group  that  contains  the  leading  note  of  the  key 
to  which  it  is  desired  to  go,  and  resolve  it  as  a  dominant  harmony. 

Second  way,  take  the  group  containing  the  keynote  of  the  key  to 
which  it  is  desired  to  go,  and  resolve  it  as  a  supertonic  harmony. 

It  is  a  good  practice,  unfortunately  not  so  much  observed  as  it 
ought  to  be,  to  write  the  diminished  seventh  chord,  when  used  for 
Transition,  as  it  would  be  in  the  key  in  which  it  is  resolved. 

r}'  I  ,£  I   Jt  I  2J  id     I  3-  I     I      I  4-  1    I       I5-|,I     I  6.  ,        I  I 

,  '-*y<-'\'  !"^  "]''T-y  rr'|| 


I.  C  to  E§  by  dominant.  2.  C  to  E$  by  supertonic.  3.  C  to  Av, 
dominant.  4.  C  to  Z>b,  supertonic.  5.  C  to  Z?b,  dominant.  6.  C 
to  supertonic.  Observe  that  in  each  case  the  diminished  seventh 
group  is  not  written  as  it  would  be  in  C,  but  as  it  would  be  in  the  key 
that  is  to  follow  it. 

One  great  advantage  of  writing  it  this  way  is,  that  it  simplifies  the 
reading,  because  a  musician  always  reads  the  harmony  as  well  as  the 


HARMONY. 


notes,  and  is  much  more  likely  to  trip  over  a  miswritten  passage  than 
a  difficult  one. 

The  only  way  to  master  this  protean  chord  is  to  write,  or  better 
still,  play  on  the  piano  all  the  changes  indicated  by  the  foregoing 
remarks. 

The  diminished  seventh  may  be  changed  into  a  group  of  first, 
third,  fifth,  and  seventh  in  two  ways.  First,  since  any  one  of  the 
notes  may  be  considered  as  the  ninth  by  lowering  it  half  a  tone,  the 
note  to  which  it  is  lowered  becomes  the  root. 

i.  2.  3.  4- 


1.  Z?l?,  lowered  diatonically  to  C. 

2.  B  lowered  diatonically  to  A  ;  enharmonically  changed  to 

3.  G,  lowered  chromatically  to  G\> ;  E,  enharmonically  changed 
to  FV. 

4.  E,  lowered  chromatically  to  E\?. 

Second  way,  by  retaining  one  note  and  moving  the  rest  up  a  half- 
tone, the  note  retained  becomes  the  root. 


In  both  these  cases  the  note  that  becomes  the  seventh  maybe  enhar 
monically  changed  into  an  augmented  sixth. 


Last  of  all,  the  rules  given  in  Chapter  XVI.  may  be  applied  to 
this  chord,  which  may  be  written  in  any  of  its  four  ways,  as  may 
best  suit  the  succession  desired. 

Transitions  are  often  effected  by  means  of  Harmonic  Sequences. 

The  Harmonic  Sequence  consists  of  a  dominant,  or  supertonic,  or 
augmented  sixth  harmony  followed  by  its  resolution,  repeated  at 
regular  intervals  ascending  or  descending. 


142 


HARMONY. 


The  varieties  that  may  be  made  are  infinite,  but  the  following  ex- 
amples will  give  a  clear  idea  of  the  operation. 


4-.-L 


If 


E- 


a 


1.  Dominant  and  tonic;  each  a  whole-tone  below  the  last.  The 
succession  of  dominant  seventh  will  explain  this  passage. 

2.  Dominant  and  tonic  ;  each  a  half-tone  lower  than  the  last.  Each 
common  chord  bears  the  relation  of  tonic  to  the  dominant  that  pre- 
cedes it,  and  of  lowered  supertonic  to  the  one  that  follows  it. 

3.  Dominant  and  tonic;  the  minor  ninth  added;  the  roots  ascend 
by  fourths,  each  root  a  whole-tone  above  the  preceding. 

4.  Is  a  reversal  of  3.  The  roots  descend  by  fourths,  each  root  a 
whole-tone  below  the  preceding. 

All  these  sequences  may  be  inverted  or  rearranged  in  any  way. 
A  Sequence  may  take  in  three,  four,  or  even  more  chords,  as  in 
the  following  examples. 


I        I        i        I  *\ 


1.  Tonic,  augmented  sixth,  dominant;  ascends  by  whole-tones. 

2.  Tonic  and  dominant;  each  root  has  first  a  tonic,  then  a  domi- 
nant harmony. 


HARMONY. 


H3 


C~2  i — ± 


=fc=t=Fi== 


}=±=±=±==i 


1 

— 

r — f  I" — f— (— 


— P — F  F— i  F--i — 11 


g  ^2  &J^=± 


f2— <S> 


1.  Supertonic  harmony;  tonic,  dominant;  descending  by  whole- 
tones. 

2.  The  same,  with  supertonic  harmony  changed  to  augmented 
sixth. 

3.  Augmented  sixth  and  dominant,  descending  by  whole-tones; 
i.  e.,  the  roots. 

4.  Same,  descending  by  half-tones;  i.  e.,  the  roots. 

Of  course  the  sequence  may  be  stopped  at  any  point,  when  the  key 
is  reached  to  which  it  is  wished  to  make  a  transition. 

Sequences  may  be  made  by  means  of  the  second  and  third  pro- 
gressions of  the  dominant,  also  with  dominant  eleventh. 


-S»  &  (2  <2_ 


fe — frg — 


&  <S>— 


i44 


HARMONY. 


1 .  A  sequence  of  second  progressions ;  each  dominant  resolved  as 
in  a  major  key. 

2.  Same;  the  dominants  resolved  as  in  a  minor  key. 

3.  Sequence  of  third  progressions. 

1.  Sequence  with  eleventh  chords. 

2.  Same,  showing  the  sudden  transition  that  may  be  made  by  sub- 
stituting A  major  for  A  minor. 

The  pupil  should  write  and  play  these  sequences  in  all  the  keys 
repeatedly,  and  should  invent  new  ones,  make  inversions  of  them, 
and  arrange  the  chords  in  different  ways. 

If  the  plan  of  instruction  mapped  out  in  this  book  has  been  care- 
fully followed,  and  each  step  mastered  before  proceeding  to  the  next, 
above  all,  if  the  Relationship  of  Keys  making  up  the  "  Group  "  is 
well  understood,  the  pupil  will  have  a  knowledge  and  command  of 
the  resources  of  harmony  that  will  amply  repay  the  time  and  labor 
spent  in  acquiring  them. 


SUPPLEMENT. 


Tempered  Scale. 

There  is  among  musicians  such  a  vague  notion  as  to  what  is  meant 
by  the  tempered  scale,  that  it  has  been  thought  well  to  add  a  few 
explanatory  words,  avoiding  all  minute  details  as  much  as  pos- 
sible. 

To  make  all  our  chords  in  perfect  tune  it  would  be  necessary  to 
divide  the  octave  into  so  many  parts  that  the  result  would  be  an  un- 
manageable mass  of  sounds  ;  but  it  was  discovered  that  by  dividing  the 
octave  as  nearly  as  possible  into  twelve  equal  parts,  a  series  of  sounds 
was  obtained,  which  while  not  corresponding  exactly  with  the  true 
series,  was  yet  so  near  that  every  sound  in  the  series  might  be  the 
root,  third,  fifth,  seventh,  etc.,  in  some  chord,  so  nearly  in  tune  that 
the  ear  was  satisfied. 

.  The  gain  to  music  was  not  alone  in  the  simplification  of  the  scale, 
but,  what  was  of  far  greater  importance,  the  power  of  passing  at 
will  from  any  key  to  any  other,  by  taking  advantage  of  the  sounds 
they  hold  in  common, 

On  this  modern  music  is  founded.  It  is  hardly  too  much  to  say 
that  modern  music  dates  from  the  publication  of  the  "Well  Tem- 
pered Clavier." 

Practically  the  "tempering"  of  the  scale  is  secured  by  a  very  sim- 
ple means;  viz.,  by  tuning  every  fifth  slightly  flat.  This  secures  the 
twelvefold  division  of  the  octave. 

The  ratio  between  the  vibrations,  per  second,  of  a  given  note  and 
its  octave  is  i  to  2 ;  that  is,  the  octave  vibrates  twice  as  fast.  The 
ratio  between  the  vibrations,  per  second,  of  a  given  note  and  its 
fifth  is  2  to  3 ;  that  is,  the  fifth  vibrates  one-half  faster  than  the 
root. 

(145) 


146 


SUPPLEMENT. 


Now  suppose  we  begin  with  C,  and  tune  in  fifths  and  octaves  as  in 
the  following  diagram. 

Oc-  Oc-  , 

Fifth.  tave.Fifth. Fifth,  tave.  .  1  I  ,1 


I       I    ~2       3      2    4  .    J     "     ~     '  4 


3  ^6   7  ^  8   9  Y  10     116  12 


It  will  be  seen  that  by  tuning  up  twelve  fifths  and  down  six  oc- 
taves, the  octave  of  the  first  note  is  reached. 

Now,  if  we  allow  144  vibrations  to  the  starting  note,  the  octave 
must  have  2S8 ;  but  the  C  reached  by  the  tuning  given  above  will 
have  about  290,  so  that  it  is  too  sharp,  about  the  one-eighth  of  a  tone. 

Tempering  is  dividing  this  small  interval,  called  the  comma  of 
Pythagoras  from  its  reputed  discoverer,  as  equally  as  possible  among 
the  twelve  fifths.  The  result  is  that  the  octave  is  the  only  interval  in 
our  system  that  is  perfectly  in  tune. 

Below  will  be  found  the  first  six  of  the  above  sounds,  with  their 
vibration  numbers  attached.  It  is  an  easy  arithmetical  problem  to 
find  the  rest.  To  find  the  fifth  of  a  given  sound,  add  half  the  num- 
ber to  itself;  thus,  C,  144;  G,  216,  because  the  half  of  144  is  72, 
and  72  and  144  are  216.  When  the  octave  below  is  to  be  found, 
divide  the  number  by  2. 


144  Sj  »4J  ,t,X 


It  is  not  necessary  here  to  go  any  deeper  into  this  subject.  The 
student  who  wishes  to  become  thoroughly  acquainted  with  it  will 
find  it  exhaustively  treated  in  many  works  on  acoustics,  especially  in 
Helmholtz,  Tyndal,  Blaserna,  etc.  There  is  also  an  excellent  trea- 
tise in  44  Grove's  Dictionary,"  under  the  title  Temperament. 

The  regret  is  often  expressed  that  musicians  do  not  adopt  a  more 
perfect  scale,  but  it  should  not  be  forgotten  that  custom  rules  in  this 


SUPPLEMENT. 


as  in  many  other  things.  From  our  earliest  infancy  we  are  habit- 
uated to  the  tempered  scale. 

It  is  the  scale  that  has  given  us  all  the  greatest  music  we  possess; 
it  satisfied  the  musical  instincts  of  Bach,  Haydn,  Mozart,  Beethoven, 
Mendelssohn,  and  countless  others  among  the  greatest  composers. 

Music  means  something  more  than  mere  sweet  sound.  To  one 
who  feels  this  meaning,  no  possible  exactness  of  intonation  would 
add  to  it  in,  say,  the  slow  movement  of  the  Sonata  Path^tique  (I 
choose  an  illustration  from  piano  music  because  the  chief  scorn  of 
the  purists  is  directed  against  the  piano),  while  on  the  other  hand  it 
would  quite  destroy  the  unexpected  modulation  from  A\>  to  E,  and 
back  again. 

There  is  but  one  way  in  which  a  change  in  our  scale  may  be  intro- 
duced :  some  great  composer  must  arise  who  will  show  us  that  its 
possibilities  for  expression  far  surpass  those  of  the  tempered  scale. 


FIGURED  BASS. 

Called  also  Erroneously  Thorough  Bass,  (Italian)  Basso 
Continuo,  (German)  General  Bass. 

Figured  bass  was  devised  as  a  sort  of  musical  shorthand,  by  means 
of  which  the  chord  each  bass  note  was  to  bear  was  represented 
by  figures  placed  under  or  over  the  bass.  Its  use  dates  from  about 
the  year  1600.  It  was  intended  to  serve  as  a  guide  to  the  accom- 
panist, to  whose  discretion  it  was  left  to  arrange  the  harmony  as  he 
pleased.  It  may  easily  be  guessed  that  accompaniment  did  not  have 
the  importance  in  the  estimation  of  the  old  composers  that  it  has  in 
their  modern  successors. 

By  one  of  those  strange  chances  that  so  often  happen,  figured  bass 
has  assumed  a  position  never  contemplated  by  its  inventors,  through 
its  adoption  as  the  means  of  teaching  harmony.  Its  original  purpose 
has  long  been  disused,  although  conservatism  still  demands  ''playing 


148 


SUPPLEMENT. 


from  figured  bass  "  as  one  of  the  exercises  for  candidates  for  degrees 
at  some  universities. 

It  may  be  easily  understood  that  it  is  quite  possible  for  any  one  to 
write  over  a  given  series  of  notes  the  intervals  indicated  by  the  fig- 
ures, without  having  the  least  conception  of  the  reasons  for  the 
combinations  or  for  their  successions.  Just  as  one  may  learn  the 
Greek  alphabet  so  as  to  pronounce  with  facility  the  Greek  language, 
yet  without  knowing  the  meaning  of  a  single  word. 

The  principles  upon  which  the  system  of  figured  bass  is  con- 
structed are  easily  understood,  and  to  anyone  who  has  mastered  the 
system  of  harmony  taught  in  this  book,  they  offer  no  difficulties. 

The  simple  combinations  and  successions  of  the  older  writers  may 
be  represented  with  comparatively  little  complexity,  but  the  most 
ardent  advocates  of  figured  bass  admit  that  the  amount  of  complica- 
tion made  necessary  in  the  figuring  by  the  complexity  of  modern 
music,  is  such  that  its  unravelment  becomes  a  veritable  enigma. 

The  rules  upon  which  the  system  is  based  are  as  follows  (the 
figures  indicate  the  intervals  over  the  bass): 

When  a  note  is  without  figures  it  bears  the  common  chord. 

1.  But  it  is  sometimes  necessary,  as  for  example,  when  a  given 
note  is  to  bear  two  or  more  chords,  to  indicate  the  common  chord  by 

8 

a  3,  or  5,  or  6  or  5,  etc.     (As  a  general  thing,  the  3  is  sufficient.) 


2.  The  position  of  the  chord  is  left  entirely  to  the  discretion,  or 
the  reverse,  of  the  student. 

1.  2. 


1 


The  figures  must  be  w 
with  the  largest  unit  at  the 
without  regard  to  the  member  of 
the  chord  that  may  occupy  this 
position ;  thus, 


ritten  / 
top,  j 


e 


5  may  be 
3 


9* 


I 


SUPPLEMENT. 


149 


When  this  arrangement  is  de- 
parted from,  it  means  that  the 
intervals  must  be  arranged  in  the 
way  indicated  by  the  figures  ;  thus, 


i 


1 


33 


The  first  inversion  of  a  common  chord  is  figured  *j,  or  6  only ;  the 
rule  being  that  where  6  is  used  3  is  understood  to  accompany  it  (i)» 
The  second  inversion  is  figured  J  (2). 
1. 


2. 


1 


— &- 


3  or  6 


l 


The  figure  7  (for  5)  indicates  any  of  the  following  groups  : 

3 

Dominant  or  supertonic,  first  to  seventh,  third  to  ninth,  or  fifth  to 
eleventh,  or  a  passing  seventh,  or  a  suspended  seventh  (1). 


-(2- 


X 


2 


-1— 


7  . 


The  first  inversion  of  5  is  figured  5,  a  3  being  understood  (1)  ;  the 

7 

seconl  inversion  of  5  is  figured  |,  a  6  being  understood  (2)  ;  the 

3 

third  inversion  of  5  is  figured  *  a  6  being  understood  (3) . 


(As  in  the  former  case  it  applies  to  any  four  note  group.) 
1.  2.  3. 


^  


 a. 




5 
3 

6 

5  or 
3 


4  or 


4  or 
2 


8 


9i 


1 


SUPPLEMENT. 


Accidentals- are  indicated  as  follows: 

When  sharp,  flat,  or  natural  is  placed  over  a  note  not  followed  by 
a  figure,  it  always  refers  to  the  third  over  the  bass  (i).  When  sharp, 
flat,  or  natural  is  followed  by  a  figure,  it  affects  the  member  of  the 
chord  for  which  that  figure  stands  (2) 


4 

}— *  q 

F 

6                      4  % 

1 

There  are  two  exceptions,  viz.,  the  augmented  sixth  is  indicated  by 
a  six  with  a  line  through  it,  thus,  the  augmented  fourth  is  indi- 
cated in  the  same  way, 

When  a  note  is  common  to  two  or  more  chords  in  succession,  its 
repetition  is  indicated  by  means  of  short  horizontal  lines  (1). 

A  long  line  placed  after  the  figures,  over  the  first  note  of  a  running 
or  arpeggioed  bass,  means  that  the  chord  is  to  be  held  until  the  end 
of  the  line  (2). 


When  there  are  several  sets  of  figures  over  a  single  note,  care  must 
be  taken  to  get  the  value  of  the  notes  in  which  the  chords  are  written 
correctly.  When  there  are  two  or  four  sets  of  figures  over  a  note, 
there  is  no  difficulty;  but  when  there  are  three  or  any  other  odd 
number,  it  is  not  so  easy  to  tell  just  how  the  values  are  to  be  dis- 


SUPPLEMENT. 


trihuted.  This  is  indicated  in  some  degree  by.  the  position  of  the 
figures  in  the  measure  ;  thus, 


Various  attempts  have  been  made  to  improve  on  this  system  of 
figured  bass  by  the  use  of  additional  signs,  but  the  increase  in  the 
number  of  signs  only  increases  the  complexity. 

One  plan  largely  adopted,  is  to  indicate  the  degree  of  the  scale  that 
is  the  root  of  the  chord  by  means  of  Roman  numerals  under  the 
chords. 

Thus,  3  signifies  that  the  chord  is  an  inversion  of  the  chord  whose 
V 

root  is  the  fifth  of  the  scale. 

Another  plan  that  has  never  become  general  uses  the  letters  of  the 
alphabet  to  indicate  the  member  of  the  chord  that  is  used  as  a  bass ; 
thus, 

A  signifies  that  the  root  is  at  the  bass. 
B  signifies  that  the  third  is  at  the  bass. 
C  signifies  that  the  fifth  is  at  the  bass. 
D  signifies  that  the  seventh  is  at  the  bass. 
E  signifies  that  the  ninth  is  at  the  bass. 
F  signifies  that  the  eleventh  is  at  the  bass. 
G  signifies  that  the  thirteenth  is  at  the  bass. 


SUPPLEMENT. 


—Q- — <s>  ^>  (9—  :  5^  a — i  n 


 c>  

 ^  

— H>« 

 ^  

 17^ 

A 

7 

B 
6 
5 

C 
4 
3 

D 
4 

2 

E 

F 

h 

2 

G 
5 
3 

Enough  has  been  written  to  give  the  student  a  thorough  under- 
standing of  the  meaning  of  Figured  Bass.  Should  any  one  wish  to 
pursue  the  subject  farther  there  are  innumerable  text-books  based  on 
this  system  that  may  be  consulted. 

That  good  musicians  may  be  trained  by  this  system  countless  num- 
bers attest.  We  only  claim  for  the  system  set  forth  here,  that  it 
reaches  the  same  results  by  a  shorter  and  pleasanter  route. 


